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i 


ELEMENTARY 
MATHEMATICAL  ANALYSIS 


McGraw-Hill  BookCompany^ 

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MODERN    MATHEMATICAL    TEXTS 

Edited  by   Charles  S.   Slighter 

ELEMENTARY 
MATHEMATICAL  ANALYSIS 

A   TEXT  BOOK  FOR  FIRST 
YEAR  COLLEGE  STUDENTS 


BY 
CHARLES  S.  SLIGHTER, 

PROFESSOR  OP  APPLIED  MATHEMATICS 
UNIVERSITY  OF  WISCONSIN 


First  Edition 


McGRAW-HILL  BOOK  COMPANY,  Inc. 

239  WEST  39TH  STREET,  NEW  YORK 

6  BOUVERIE  STREET,   LONDON.  E.  C. 
1914 


::^ 


56 


Copyright,  1914,  by  the 
McGraw-Hill  Book  Company,  Inc. 


THE.  MAPLE.  PRESS.  TOKK.  PA 


PREFACE 


This  book  is  not  intended  to  be  a  text  on  "Practical  Mathe- 
matics" in  the  sense  of  making  use  of  scientific  material  and  of 
fundamental  notions  not  already  in  the  possession  of  the  student, 
or  in  the  sense  of  making  the  principles  of  mathematics  secondary 
to  its  technique.  On  the  contrary,  it  has  been  the  aim  to  give 
the  fundamental  truths  of  elementary  analysis  as  much  prominence 
as  seems  possible  in  a  working  course  for  freshmen. 

The  emphasis  of  the  book  is  intended  to  be  upon  the  notion  of 
functionality.  Illustrations  from  science  are  freely  used  to  make 
this  concept  prominent.  The  student  should  learn  early  in  his 
course  that  an  important  purpose  of  mathematics  is  to  express  and 
to  interpret  the  laws  of  actual  phenomena  and  not  primarily  to 
secure  here  and  there  certain  computed  results.  Mathematics 
might  well  be  defined  as  the  science  that  takes  the  broadest  view  of 
all  of  the  sciences — an  epitome  of  quantitative  knowledge.  The 
introduction  of  the  student  to  a  broad  view  of  mathematics  can 
hardly  begin  too  early. 

The  ideas  explained  above  are  developed  in  accordance  with  a 
two-fold  plan,  as  follows: 

First,  the  plan  is  to  group  the  material  of  elementary  analysis 
about  the  consideration  of  the  three  fundamental  functions : 

1.  The  Power  Function  y  =  ax"  (n  any  number)  or  the  law 
"as  X  changes  by  a  fixed  multiple,  y  changes  hy  a  fixed  multiple  also.^' 

2.  The  Simple  Periodic  Function  y  =  a  sin  mx,  considered  as 
fundamental  to  all  periodic  phenomena. 

3.  The  Exponential  Function,  or  the  law  "asx  changes  hy  a  fixed 
increment,  y  changes  by  a  fixed  multiple.^' 

Second,  the  plan  is  to  enlarge  the  elementary  functions  by  the 
development  of  the  fundamental  transformations  applicable  to 


2J)22«^ 


vi  PREFACE 

these  and  other  functions.  To  avoid  the  appearance  of  abstruse- 
ness,  these  transformations  are  stated  with  respect  to  the  graphs 
of  the  functions;  that  is,  they  are  not  called  transformations,  but 
"  motions  "  of  the  loci.  The  facts  are  summarized  in  several  simple 
''Theorems  on  Loci,"  which  explain  the  translation,  rotation,  shear, 
and  elongation  or  contraction  of  the  graph  of  any  function  in  the 
xy  plane. 

Combinations  of  the  fundamental  functions  as  they  actually 
occur  in  the  expression  of  elementary  natural  laws  are  also  dis- 
cussed and  examples  are  given  of  a  type  that  should  help  to  explain 
their  usefulness. 

Emphasis  is  placed  upon  the  use  of  time  as  a  variable.  This 
enriches  the  treatment  of  the  elementary  functions  and  brings 
many  of  the  facts  of  ''analytic  geometry"  into  close  relation  to 
their  application  in  science.  A  chapter  on  waves  is  intended  to 
give  the  student  a  broad  view  of  the  use  of  the  trigonometric  func- 
tions and  an  introduction  to  the  application  of  analysis  to  peri- 
odic phenomena. 

It  is  difficult  to  understand  why  it  is  customary  to  introduce 
the  trigonometric  functions  to  students  seventeen  or  eighteen  years 
of  age  by  means  of  the  restricted  definitions  applicable  only  to  the 
right  triangle.  Actual  test  shows  that  such  rudimentary  methods 
are  wasteful  of  time  and  actually  confirm  the  student  in  narrow- 
ness of  view  and  in  lack  of  scientific  imagination.  For  that  reason, 
the  definitions,  theorems  arid  addition  formulas  of  trigonometry 
are  kept  as  general  as  practicable  and  the  formulas  are  given 
general  demonstrations. 

The  possibilities  and  responsibilities  of  character  building  in  the 
department  of  mathematics  are  kept  constantly  in  mind.  It  is 
accepted  as  fundamental  that  a  modern  working  course  in  mathe- 
matics should  emphasize  proper  habits  of  work  as  well  as  proper 
methods  of  thought;  that  neatness,  system,  and  orderly  habits 
have  a  high  value  to  all  students  of  the  sciences,  and  that  a  text- 
book should  help  the  teacher  in  every  known  way  to  develop  these 
in  the  student. 

Chapters  V,  VI  and  VII  cpntain  material  that  is  required  for 
admission  to  many  colleges  and  universities.  The  amount  of  time 
devoted  to  these  chapters  will  depend,  of  course,  upon  the  local 
requirements  for  admission. 


PREFACE  VI 1 

The  present  work  is  a  revision  and  rewriting  of  a  preliminary 
form  which  has  been  in  use  for  three  years  at  the  University  of 
Wisconsin.  During  this  time  the  writer  has  had  frequent  and 
valuable  assistance  from  the  instructional  force  of  the  department 
of  mathematics  in  the  revision  and  betterment  of  the  text.  Ac- 
knowledgments are  due  especially  to  Professors  Burgess,  Dresden, 
Hart  and  Wolff  and  to  Instructors  Fry,  Nyberg  and  Taylor. 
Professor  Burgess  has  tested  the  text  in  correspondence  courses, 
and  has  kindly  embraced  that  opportunity  to  aid  very  materially 
in  the  revision.  He  has  been  especially  successful  in  shortening 
graphical  methods  and  in  adapting  them  to  work  on  squared  paper. 
Professor  Wolff  has  read  all  of  the  final  manuscript  and  made 
many  suggestions  based  upon  the  use  of  the  text  in  the  class  room. 
Mr.  Taylor  has  read  all  of  the  proof  and  supplied  the  results  to  the 
exercises. 

Professor  E.  V.  Huntington  of  Harvard  University  has  read  the 
galley  proof  and  has  contributed  many  important  suggestions. 

The  writer  has  avoided  the  introduction  of  new  technical  terms, 
or  terms  used  in  an  unusual  sense.  He  has  taken  the  liberty,  how- 
ever of  naming  the  function  ax",  the  "Power  Function  of  a:,"  as  a 
short  name  for  this  important  function  seems  to  be  an  unfortu- 
nate lack — a  lack,  which  is  apparently  confined  solely  to  the 
English  language. ' 

It  is  with  hesitation  that  the  writer  acknowledges  his  indebted- 
ness to  the  movement  for  the  improvement  of  mathematical  in- 
struction that  has  been  led  by  Professor  Klein  of  Gottingen; 
not  that  this  is  not  an  attempt  to  produce  a  text  in  harmony  with 
that  movement,  but  for  fear  that  the-  interpretation  expressed 
by  the  present  book  is  inadequate. 

The  writer  will  be  glad  to  receive  suggestions  from  those  that 
make  use  of  the  text  in  the  class  room. 

Charles  S.  Slighter. 
University  of  Wisconsin 
July,  25,  1914 


CONTENTS 

Preface v 

Introduction xi 

Mathematical  Signs  and  Symbols xiv. 

Chapter 

I.  Variables  and  Functions  of  Variables 1 

II.  Rectangular    Coordinates    and  the   Power   Func- 
tion    25 

III.  The  Circle  and  the  Circle  Functions 94 

IV.  The  Ellipse  and  Hyperbola 137 

V.  Single  and  Simultaneous  Equations 162 

VI.  Permutations,    Combinations,    the    Binomial 

Theorem 182 

VII.  Progressions 198 

VIII.  The  Logarithmic  and  Exponential  Functions  .    .    .  214 
IX.  Trigonometric    Equations    and    the    Solution    of 

Triangles 282 

X.  Waves 321 

XI.  Complex  Numbers 341 

XII.  Loci     381 

XIII.  The  Conic  Sections 398 

XIV.  A  p  p  E  n  D I X — A    Review    of    Secondary    School 

Algebra 452 

Index 483 

ix 


INTRODUCTION 

Any  course  in  mathematics  requires  the  frequent  use  of 
geometrical  constructions,  and  the  carrying  out  of  analytical 
and  numerical  computations.  In  order  that  this  work  may 
be  performed  neatly  and  accurately  it  is  necessary  that  the 
student  have  a  few  simple  instruments,  and  a  supply  of  proper 
material  for  doing  the  work  in  a  systematic  and  orderly  manner. 
The  indispensible  instruments  are  as  follows: 

I.  Instruments.  (1)  Two  AH  hexagonal  drawing  pencils;  one 
sharpened  to  a  fine  point  for  marking  points  upon  paper  or 
for  sketching  free  hand;  the  other  sharpened  to  a  chisel  point 
for  drawing  straight  lines.  Some  prefer  to  use  a  single  pencil 
sharpened  at  both  ends,  one  end  round  pointed,  the  other  end 
chisel  pointed. 

(2)  A  small  drawing  board^  of  soft  wood — 10  X  12  inches  is 
large  enough. 

(3)  A  small  T-square,  same  length  as  the  drawing  board. 

(4)  A  60°  and  a  45°  transparent  triangle.  Five-inch  triangles 
are  large  enough,  although  a  larger  60°  triangle  will  be  found  to 
be  very  convenient. 

(5)  A  protractor  for  laying  off  angles. 

(6)  A  triangular  boxwood  scale,  decimally  divided. 

(7)  A  pair  of  6-inch  pencil  compasses  for  drawing  circles  and 
arcs  of  circles,  provided  with  medium  hard  lead,  sharpened  to  a 
narrow  chisel  point. 

(8)  A  10-inch  slide  rule  is  required  for  Chapter  VIII,  and  may 
be  used  earlier  at  the  discretion  of  the  instructor. 

1  Drawing  boards  of  this  size  with  T-square  and  two  wood  triangles  are  marketed 
by  the  Milton  Bradly  Co.,  Springfield  Mass.,  and  by  Eugene  Dietzgen  Co.,  and 
Keuffel  and  Esser  of  New  York  and  Chicago,  and  retail  for  about  40  cents. 

xi 


xii  INTRODUCTION 

II.  Materials.  All  mathematical  work  should  be  done  on  one 
side  of  standard  size  letter  paper,  8|  X  11  inches.  This  is  the 
smallest  sheet  that  permits  proper  arrangement  of  mathematical 
work.     There  are  required: 

(1)  A  note  book  cover  to  hold  sheets  of  the  above  named  size  and 
a  supply  of  manila  paper  ''vertical  file  folders"  for  use  in  submit- 
ting work  for  the  examination  of  the  instructor. 

(2)  A  number  of  different  forms  of  squared  paper  and  computa- 
tion paper  especially  prepared  for  use  with  this  book.  These  sheets 
will  be  described  from  time  to  time  as  needed  in  the  work.  Form 
M2  will  be  found  convenient  for  problem  work  and  for  general 
calculation.  M2  is  a  copy  of  a  form  used  by  a  number  of  public 
utility  and  industrial  corporations.  Colleges  usually  have  their 
own  sources  of  supply  of  squared  paper,  satisfactory  for  use  with 
this  book.  The  forms  mentioned  in  the  text,  printed  on  16  lb., 
St.  Regis  Bond,  cost  about  25  cents  per  pound  in  100  lb.  lots 
(12,000  sheets)  from  F.  C.  Blied  &  Co.,  Madison,  Wis. 

(3)  Miscellaneous  supplies  such  as  thumb  tacks,  erasers,  sand- 
paper-pencil-sharpeners, etc. 

III.  General  Directions.  All  drawings  should  be  done  in 
pencil,  unless  the  student  has  had  training  in  the  use  of  the  ruling 
pen,  in  which  case  he  may,  if  he  desires,  "ink  in"  the  most  im- 
portant drawings. 

All  mathematical  work,  such  as  the  solutions  of  problems  and 
exercises,  and  work  in  computation  should  be  done  in  ink.  The 
student  should  acquire  the  habit  of  working  problems  with  pen 
and  ink.  He  will  find  that  this  habit  will  materially  aid  him  in 
repressing  carelessness  and  indifference  and  in  acquiring  neatness 
and  system. 

TO  THE  INSTRUCTOR 

The  usual  one  and  one-half  year  of  secondary  school  Algebra 
including  the  solution  of  quadratic  equations  and  a  knowledge  of 
fractional  and  negative  exponents,  is  required  for  the  work  of  this 
course.  In  the  appendix  will  be  found  material  for  a  brief  review 
of  factoring,  quadratics,  and  exponents,  upon  which  a  week  or  ten 
days  should  be  spent  before  beginning  the  regular  work  in  this 
text. 


INTRODUCTION 


Xlll 


The  instructor  cannot  insist  too  emphatically  upon  the  require- 
ment that  all  mathematical  work  done  by  the  student — whether 
preliminary  work,  numerical  scratch  work,  or  any  other  kind 
(except  drawings) — shall  be  carried  out  with  pen  and  ink  upon 
paper  of  suitable  size.  This  should,  of  course,  include  all  work' 
done  at  home,  irrespective  of  whether  it  is  to  be  submitted  to  the 
instructor  or  not.  The  "psychological  effect"  of  this  requirement 
will  be  found  to  entrain  much  more  than  the  acquirement  of  mere 
technique.  If  properly  insisted  upon,  orderly  and  systematic 
habits  of  work  will  lead  to  orderly  and  systematic  habits  of 
thought.  The  final  results  will  be  very  gratifying  to  those  who 
sufficiently  persist  in  this  requirement. 

At  institutions  whose  requirements  for  admission  include  more 
than  one  and  one-half  units  of  preparatory  algebra,  nearly  all  of 
Chapters  V,  VI,  and  VII  may  be  omitted  from  the  course. 

An  asterisk  attached  to  a  section  number  indicates  that  the 
section  may  he  omitted  during  the  first  reading  of  the  book. 

GREEK  ALPHABET 


Capitals 

Lower 
case 

Names 

Capitals 

Lower 
case 

Names 

A 

a 

Alpha 

N 

V 

Nu 

B 

/3 

Beta 

^ 

Xi 

r 

T 

Gamma 

0 

o 

Omicron 

A 

5 

Delta 

II 

T 

Pi 

E 

e 

Epsiloii 

P 

P 

Rho 

Z 

f 

Zeta 

s 

a 

Sigma 

H 

V 

Eta 

T 

T 

Tau 

0 

6 

Theta 

T 

V 

Upsilon 

I 

t 

Iota 

* 

4> 

Phi 

K 

K 

Kappa 

X 

X 

Chi 

A 

X 

Lambda 

^ 

^ 

Psi 

M 

M 

Mu 

fi 

CO 

Omega 

XIV  INTRODUCTION 


... 

read 

= 

read 

5^ 

read 

= 

read 

= 

read 

> 

read 

< 

read 

> 

read 

{a,h) 

read 

\n 

read 

n\ 

read 

lim      .,  . 

read 

x^  a 

X  =    CO 

read 

\a\ 

read 

lOgaX 

read 

hx 

read 

In  X 

read 

n  =.  r 

read 

MATHEMATICAL  SIGNS  AND  SYMBOLS 


and  so  on. 

is  identical  with. 

is  not  equal  to. 

approaches. 

is  approximately  equal  to. 

is  greater  than. 

is  less  than. 

is  greater  than  or  equal  to. 

point  whose  coordinates  are  a  and  h. 

factorial  n. 

factorial  n  or  n  admiration. 

limit  of  f{x)  as  x  approaches  a. 

x  becomes  infinite, 
absolute  value  of  a. 
logarithm  of  x  to  the  base  a. 
common  logarithm  of  x. 
natural  logarithm  of  x. 

summation  from  n  =  1  to  n  =  t  of  Un. 


ELEMENTARY 
MATHEMATICAL  ANALYSIS 

CHAPTER  I 

VARIABLES  AND  FUNCTIONS  OF  VARIABLES 

1.  Scales.  If  a  series  of  points  corresponding  in  order  to  the 
numbers  of  any  sequence^  be  selected  along  any  curve,  the  curve 
with  its  points  of  division  is  called  a  scale.  Thus  in  Fig.  1  (a) 
the  points  along  the  curve  OA  have  been  selected  and  marked  in 
order  with  the  numbers  of  the  sequence: 

0,1/4,1/2,1,21,3,5,7,8 

Thus  primitive  man  might  have  made  notches  along  a  twig 
and  then  made  use  of  it  in  making  certain  measurements  of 


^  1  ""-'^^  i^^  7 


0 

(a)   A  Non  L'niform  Scale 

li  1 1  I  li  I  I  I  h  M  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  1 1  I  I  I  h  I  I  I  h  I  I  1 1  II I  1 1 

0  12  3  4  5 

(b)  A  Uniform  Arithmetical  Scale 

1  '    I   I    I    I   I   I    I   I    I  I    I    I    I  h    I   I    I   I    I    I   II   I   I    II   I  I   I   II   I    I  II    II    I  I   M    I  I    I   M    I    I 

-5       -4        -S       -2        .1  0        +1       +3       +3       +4+5 

(c)  A  Uniform  Algebraic  Scale 

Fig.   1. — Scales  of  Various  Sorts. 

interest  to  him.  If  such  a  scale  were  to  become  generally  used  by 
others,  it  would  be  desirable  to  make  many  copies  of  the  original 
scale.  It  would,  therefore,  be  necessary  to  use  a  twig  whose  shape 
could  be  readily  duplicated;  such,  for  example,  as  a  straight  stick; 
and  it  would  also  be  necessary  to  attach  the  same  symbols  in- 
variably to  the  same  divisions. 

^  A  sequence  of  numbers  here   means  a  set  of  numbers   arranged  in  order  of 
magnitude. 

1 


.  EI^EMBNTARY  MATIIl^MATICAL  ANALYSIS 


[§1 


Certain  advantages  are  gained  (often  at  the  expense  of  others, 
however)  if  the  distances  between  consecutive  points  of  division 
are  kept  the  same;  that  is,  when  the  intervals  are  laid  off  by  repe- 
tition of  the  same  selected  distance.     When  this  is  done,  the  scale 


Fig.  2. — An  Ammeter  Scale. 


is  called  a  uniform  scale.  Primitive  man  might  have  selected  for 
such  uniform  distance  the  length  of  his  foot,  or  sandal,  the  breadth 
of  his  hand,  the  distance  from  elbow  to  the  end  of  the  middle 
finger  (the  cubit),  the  length  of  a  step  in  pacing  (the  yard),  the 

amount  he  can  stretch  with  both 
arms  extended  (the  fathom),  etc., 
etc. 

We  are  famiUar  with  many 
scales,  such  as  those  seen  on  a 
yardstick,  the  dial  of  a  clock,  a 
thermometer,  a  sun-dial,  a  steam- 


FlG. 


1    12  U 
3. — Sun-dial  Scale 


gage,    an    ammeter    or  voltmeter, 

the  arm  of  a  store-keeper's  scales, 

etc.,  etc.     The  scales  on  a  clock,  a  yardstick,  or  a  steel  tape  are 

uniform.     Those  on  a  sun-dial,  on  an  ammeter  or  on  a  good 

thermometer,  are  not  uniform. 

One  of  the  most  important  advantages  of  a  uniform  scale  is 
the  fact  that  the  place  of  beginning  or  zero  may  be  taken  at  any 
one  of  the  points  of  division.     This  is  not  true  of  a  non-uniform 


§2]        VARIABLES  AND  FUNCTIONS  OF  VARIABLES  3 

scale.  If  the  needle  of  an  ammeter  be  bent  the  instrument  cannot 
be  used.  It  is  always  necessary  in  using  such  an  instrument  to 
know  that  the  zero  is  correct;  if  a  sun-dial  is  not  properly  oriented, 
it  is  useless.  If,  however,  a  yardstick  or  a  steel  tape  be  broken, 
it  may  still  be  used  in  measuring.  The  student  may  think  of 
many  other  advantages  gained  in  using  a  uniform  scale. 

2.  Formal  Definition  of  a  Scale.  If  points  be  selected  in  order 
along  any  curve  corresponding,  one  to  one,  to  the  numbers  of 
any  sequence,  the  curve,  with  its  divisions,  is  called  a  scale. 

The  notion  of  one  to  one  correspondence,  included  in  this 
definition,  is  frequently  used  in  mathematics. 

In  mathematics  we  frequently  speak  of  the  arithmetical  scale 
and  of  the  algebraic  scale.  The  arithmetical  scale  corresponds  to 
the  numbers  of  the  sequence: 

0,  1,  2,  3,  4,  5,  .  .  . 

and  such  intermediate  numbers  as  may  be  desired.  It  is 
usually  represented  by  a  uniform  scale  as  in  Fig.  1  (6).  The 
algebraic  scale  corresponds  to  the  numbers  of  the  sequence: 

.    .    .    -6, -5, -4, -3,  -2,-1,0, +1, +2, +3, +4, +5,  .    .    . 

and  such  intermediate  numbers  as  may  be  desired.  It  is  usually 
represented  by  a  uniform  scale  as  in  Fig.  1  (c).  The  arithmetical 
scale  begins  at  0  and  extends  indefinitely  in  one  direction.  The 
algebraic  scale  has  no  point  of  beginning;  the  zero  is  placed  at  any 
desired  point  and  the  positive  and  negative  numbers  are  then 
attached  to  the  divisions  to  the  right  and  the  left,  respectively,  of 
the  zero  so  selected.  The  scale  extends  indefinitely  in  both 
directions. 

Exercises 

1.  Show  that  the  distance  between  two  points  selected  anywhern 
on  the  algebraic  scale  is  always  found  by  subtraction. 

2.  If  two  algebraic  scales  intersect  at  right  angles,  the  commoe 
point  being  the  zero  of  both  scales,  explain  how  to  find  the  distance 
from  any  point  of  one  scale  to  any  point  of  the  other  scale. 

3.  What  points  of  the  algebraic  scale  are  distant  5  from  the 
point  3  of  that  scale?  What  point  of  the  arithmetical  scale  is 
distant  5  from  the  point  3  of  that  scale? 


ELEMENTARY  MATHEMATICAL  ANALYSIS 


(§2 


-~-^ 


■a  s- 

o 


^     2 


0)      o 


m 


VARIABLES  AND  FUNCTIONS  OF  VARIABLES 


3.  Two  Scales  in  Juxtaposition  or  Double  Scales.  The  relation 
between  two  magnitudes  or  quantities,  or  between  two  numbers, 
may  be  shown  conveniently  by  placing  two  scales  side  by  side. 
Thus  the  relation  between  the  number  of  centimeters  and  the 
number  of  inches  in  any  length  may  be  shown  by  placing  a  centi- 
meter scale  and  afoot-rule  side 
by  side  with  their  zeros  coin- 
ciding as  in  Fig.  4. 

A  thermometer  is  frequent- 
ly seen  bearing  both  the 
Fahrenheit  and  the  centi- 
grade scales  (see  Fig.  5). 
It  is  obvious  that  the  double 
scale  of  such  a  thermometer 
may  be  used  (within  the 
limits  of  its  range)  for  convert- 
ing any  temperature  reading 
Fahrenheit  into  the  corres- 
ponding centigrade  equiva- 
lent and  vice  versa.  The  con- 
struction of  scales  of  this 
sort  may  be  made  to  depend 
upon  the  solution  of  the  fol- 
lowing problem  in  elementary 
geometry:  To  divide  a  given 
line  into  a  given  number  of  equal  parts. 

To  construct  a  double  scale  showing  the  relation  between  speed 
expressed  in  miles  per  hour,  and  speed  expressed  in  feet  per  second, 
we  may  proceed  as  follows :  A  mile  contains  5280  feet;  an  hour  con- 
tains 3600  seconds.  Hence,  one  mile  per  hour  equals  5280/3600 
or  22  /15  feet  per  second.  On  one  of  two  intersecting  straight 
lines,  OA  (see  Fig.  7),  lay  off  22  convenient  equal  intervals  (say  1  /4 
inch  each).  On  the  second  of  the  intersecting  Unes,  OB,  lay  off 
15  equal  intervals  (say  1/2  inch  each).  Join  the  15th  division 
of  OB  with  the  22nd  division  of  OA  and  draw  parallels  to  the 
line  AB  through  each  of  the  15  divisions  of  OB.  Then  the  22  and 
the  15  equal  subdivisions  stand  in  juxtaposition  along  OA  and 
constitute  the  double  scale  required.    Labelling  the  first  scale 


o 

Fig.  7. — Method  of  Construction 
of  Double  Scale  showing  Relation 
between  "  Miles  per  Hour"  and  "Feet 
per  Second." 


6  ELEMENTARY  MATHEMATICAL  ANALYSIS  [§3 

'^feet  per  second"  and  the  second  scale  ''miles  per  hour,"  the 
double  scale  may  be  used  for  converting  speed  expressed  in 
either  unit  into  speed  expressed  in  the  other. 

By  annexing  the  appropriate  number  of  ciphers  to  the  numbers 
of  each  scale,  the  range  of  the  double  scale  may  be  considered 
220  and  150  or  2200  and  1500,  etc.,  respectively. 

The  lengths  of  the  various  units  selected  for  the  diagram  are,  of 
course,  arbitrary.  As,  however,  the  student  is  expected  to  prepare 
the  various  constructions  and  diagrams  required  for  the  exercises 
in  this  book  on  paper  of  standard  letter  size  (that  is,  8|  by  11 
inches),  the  various  units  selected  should  be  such  as  to  permit  a 
convenient  and  practical  construction  upon  sheets  of  that  size. 


Exercises 

The  student  is  expected  to  carry  out  the  actual  construction  of  only 
two  of  the  double  or  triple  scales  described  in  the  following  exercises. 

1.  Construct  a  double  scale  ten  inches  long  expressing  the  relation 
between  fractions  of  an  inch  expressed  in  tenths  and  fractions  of  an 
inch  expressed  in  sixteenths. 

To  draw  this  double  scale  it  is  merely  necessary  to  lay  off  the 
intervals  directly  from  suitable  foot-rules.  On  the  scale  of  tenths 
indicate  the  inch  and  half  inch  intervals  by  longer  division  lines  than 
the  others.  On  the  scale  of  sixteenths  represent  the  quarter  inch  inter- 
vals by  longer  division  lines  than  those  of  the  sixteenths,  and  represent 
the  half  inch  and  inch  intervals  by  still  longer  lines,  as  is  usually  done 
on  foot  rules. 

2.  Draw  a  double  scale  showing  pressure  expressed  as  inches  of 
mercury  and  as  feet  of  water,  knowing  that  the  density  of  mercury 
is  13.6  times  that  of  water. 

These  are  two  of  the  common  ways  of  expressing  pressure.  Water 
pressure  at  water  power  plants,  and  often  for  city  water  service,  is 
expressed  in  terms  of  head  in  feet.  Barometric  pressure,  and  the 
vacuum  in  the  suction  pipe  of  a  pump  and  in  the  exhaust  of  a  con- 
densing steam  engine  are  expressed  in  inches  of  mercury.  The 
approximate  relations  between  these  units,  i.e.,  1  atmosphere  =  30 
inches  of  mercury  =  32  feet  of  water  =  15  pounds  per  square  inch, 
are  known  to  every  student  of  elementary  physics.  To  obtain,  in 
terms  of  feet  of  water,  the  pressure  equivalent  of  1  foot  of  mercury, 
the  latter  must  be  multiplied  by  13.6,  the  density  of  mercury.     This 


§3]        VARIABLES  AND  FUNCTIONS  OF  VARIABLES  7 

result  when  divided  by  12  gives  the  pressure  equivalent  of  1  inch  of 
mercury,  which  is  1.13  feet  of  water. 

If  we  let  the  scale  of  inches  of  mercury  range  from  0  to  10,  then  the 
scale  of  feet  of  water  must  range  from  0  to  11.3.  Hence  draw  a  line 
OA  10  inches  long  divided  into  inches  and  tenths  to  represent  inches 
of  mercury.  Draw  any  line  OB  through  0  and  lay  off  11.3  uniform 
intervals  (inch  intervals  will  be  satisfactory)  on  OB.  Connect  the 
end  division  on  OA  with  the  end  division  on  OB  by  a  line  AB.  Then 
from  1,  2,  3,  .  .  .  inches  on  OB  draw  parallels  to  BA,  thus  forming 
adjacent  to  OA  the  scale  of  equivalent  feet  of  water.  Each  of  these 
intervals  can  then  be  subdivided  into  10  equal  parts  corresponding 
to  tenths  of  feet  of  water. 

3.  Draw  a  triple  scale  showing  pressure  expressed  as  feet  of  water, 
as  inches  of  mercury,  and  as  pounds  per  square  inch,  knowing  that  the 
density  of  mercury  is  13.6  and  that  one  cubic  foot  of  water  weighs 
62.5  pounds. 

To  reduce  feet  of  water  to  pounds  per  square  inch,  the  weight  of  one 
cubic  foot  of  water,  62.5  pounds,  must  be  divided  by  144,  the  number 
of  square  inches  on  one  face  of  a  cubic  foot.  This  gives  1  foot  of 
water  equivalent  to  62.5/144  or  0.434  pounds  per  square  inch.  To 
obtain  the  pressure  given  by  1  foot  of  mercury,  the  pressure  equiva- 
lent of  1  foot  of  water  must  be  multiplied  by  13.6,  the  density  of 
mercury.  This  result  when  divided  by  12  gives  the  pressure  equiva- 
lent of  1  inch  of  mercury,  or  0.492  pounds  per  square  inch. 

One  pound  per  square  inch  is  equivalent,  therefore,  to  1/0.434  or 
2.30  feet  of  water  or  to  1/0.492  or  2.03  inches  of  mercury.  If  we  let 
the  scale  of  pounds  range  from  0  to  10,  we  may  select  1  inch  as  the 
equivalent  of  1  pound  per  square  inch,  and  divide  the  scale  OA  into 
inches  and  tenths  to  represent  this  magnitude.  Draw  two  inter- 
secting lines  OB  and  OC  through  0,  and  lay  off  23  uniform  intervals 
on  OB  and  lay  off  20.3  uniform  intervals  on  OC,  1/2  inch  being  a 
convenient  length  for  each  of  these  parts.  Connect  the  end  divisions 
of  OB  and  OC  with  A  and  through  all  points  of  division  of  OB  draw 
lines  parallel  to  BA  and  through  all  points  of  division  of  OC  draw  lines 
parallel  to  CA,  and  subdivide  into  halves  the  intervals  of  the  scales 
last  drawn.  The  range  may  be  extended  to  any  amount  desired  by 
annexing  ciphers  to  the  numbers  attached  to  the  various  scales. 

Extending  the  range  by  annexing  ciphers  to  the  attached  numbers 
is  obviously  practicable  so  long  as  the  various  intervals  or  units  are 
decimally  subdivided.  The  method  is  impracticable  for  scales  that 
are  not  decimally  subdivided,  such  as  shillings  and  pence,  degrees  and 
minutes,  feet  and  inches,  etc. 


8  ELEMENTARY  MATHEMATICAL  ANALYSIS  [§3 

4.  Draw  a  triple  scale  showing  the  relations  between  the  cubic  foot, 
the  gallon  and  the  liter,  if  1  cubic  foot  =  7^  gallons  =  28i  liters. 
Divide  the  scale  of  cubic  feet  into  tenths,  the  scale  of  gallons  into 
quarts,  and  the  third  scale  into  liters. 

It  is  obvious  that  it  is  always  necessary  first  to  select  the  range  of 
the  various  scales,  but  it  is  quite  as  well  in  this  case  to  show  the  equiva- 
lents for  1  cubic  foot  only,  as  numbers  on  the  various  scales  can  be 
multiplied  by  10,  100,  or  1000,  etc.,  to  show  the  equivalents  for  larger 
amounts. 

Select  10  inches  =  1  cubic  foot  for  the  scale  (OA)  of  cubic  feet. 
Draw  two  intersecting  lines  OB  and  OC.  On  OB  lay  off  71  equal 
parts  (say,  7^  inches)  and  on  OC  lay  off  28^  equal  parts  (say,  28f 
quarter  inches).  Connect  the  end  divisions  with  A  and  draw  the 
parallel  lines  exactly  as  with  previous  examples.  The  intervals  of  the 
scale  of  gallons  can  then  be  subdivided  into  the  four  equal  parts  to 
show  quarts. 

5.  The  velocity  in  feet  per  second  of  a  falling  body  is  given  by  the 
formula  v  =  gt,  in  which  g  =  32.2  and  t  is  measured  in  seconds.  Draw 
a  double  scale  showing  the  velocity  at  any  time. 

It  is  obvious  that  the  reading  32.2  on  the  y-scale  must  be  placed 
opposite  the  mark  1  on  the  ^-scale.  First,  select  the  range  for  the 
t-scale,  say  from  1  to  10  seconds.  Then  a  convenient  scale  for  t  is  1 
inch  equals  1  second,  which  scale  can  readily  be  subdivided  to  show 
1/5  or  1/10  seconds.  If  the  general  method  be  followed,  it  would  be 
necessary  to  lay  off  322  equal  parts  on  a  line  (OB)  intersecting  the 
/-scale  (OA).  As  this  is  an  inconveniently  large  number,  it  is  better 
to  lay  off  3.22  divisions  on  the  construction  line  OB.  Each  of  these 
divisions  may  be  2  inches  in  length,  so  that  6.44  inches  will  represent 
the  terminal  or  end  division  on  the  intersecting  line  OB.  From  the 
6.44  inch  mark  on  OB  draw  a  line  to  10  on  the  ^-scale  OA.  Then  from 
2,  4,  6  inches  on  OB  draw  parallels  to  BA,  thus  locating  v  =  100,200, 
and  300.  These  intervals  can  then  be  subdivided  into  10  equal  parts 
to  show  y  =  10,  20,  30,  .  .  .  If  values  of  v  are  wanted  for  t>  10, 
zeros  may  be  annexed  to  the  numbers  attached  to  both  scales. 

6.  Select  sections  from  any  of  the  double  scales  described  above  and 
discuss  the  relation  of  the  number  of  units  on  one  side  to  the  number  of 
units  on  the  other  side.  Show  that  the  ratio  in  different  sections  of 
the  number  of  units  on  the  two  sides  of  the  same  double  scale  is  not 
constant  if  one  scale  be  a  non-uniform  scale. 

7.  If  a  double  scale  be  drawn  on  a  deformable  body,  as,  for  example, 
on  a  rubber  band,  would  the  double  scale  still  represent  true  relations 


§4]        VARIABLES  AND  FUNCTIONS  OF  VARIABLES  9 

when  the  rubber  band  is  stretched?  What  if  .the  stretching  were 
not  uniform? 

-  4.  Functions.  The  relation  between  two  magnitudes  expressed 
graphically  by  two  scales  drawn  in  juxtaposition,  as  above,  may 
sometimes  be  expressed  also  by  means  of  an  equation.  Thus, 
if  y  is  the  number  of  dollars,  and  x  is  the  number  of  pounds  sterling 
in  any  amount,  then: 

y  =  4.87  X  (1) 

also,  if  F  be  the  reading  Fahrenheit,  and  C  the  reading  centigrade 
of  any  temperature,  then: 

F  =  IC  +  32  (2) 

also, 

U  =  13.6  F/12  =  144Tf /62.5  (3) 

where  C/,  V,  and  W  are  pressures  measui*ed,  respectively,  in  feet  of 
water,  inches  of  mercury,  or  in  pounds  per  square  inch. 

Note.  The  letters  x,  y,  F,  C,  U,  V,  W  in  the  above  equations 
stand  for  numbers;  to  make  this  emphatic  we  sometimes  speak  of  them 
as  'pure  or  abstract  numbers.  These  numbers  are  thought  of  as  arising 
from  the  measurement  of  a  magnitude  or  quantity  by  the  application 
of  a  suitable  unit  of  measure.  Thus  from  the  magnitude  or  quantity 
of  water,  12  gallons,  arises,  by  use  of  the  unit  of  measure  the  gallon,  the 
abstract  number  12. 

Algebraic  equations  express  the  relation  between  numbers,  and  it 
should  always  be  understood  that  the  letters  used  in  algebra  sta7id 
for  numbers  and  not  for  quantities  or  magnitudes. 

Quantity  or  Magnitude  is  an  answer  to  the  question:  ''How 
much?"    Niunber  is  an  answer  to  the  question:     "How  many?" 

An  interesting  relation  is  given  by  the  scales  in  Fig.  6.  This 
diagram  shows  the  fee  charged  for  money  orders  of  various 
amounts;  the  amount  of  the  order  may  first  be  found  on  the  upper 
scale  and  then  the  amount  of  the  fee  may  be  read  from  the  lower 
scale.  The  relation  here  exhibited  is  quite  different  from  those 
previously  given.  For  example,  note  that  as  the  amount  of  the 
order  changes  from  $50.01  to  $60  the  fee  does  not  change,  but 
remains  fixed  at  20  cents.  Then  as  the  amount  of  the  order 
changes  from  $60.00  to  $60.01,  the  fee  changes  abruptly  from  20 
cents  to  25  cents.     For  an  order  of  any  amount  there  is  a  cor- 


10  ELEMENTARY  MATHEMATICAL  ANALYSIS  [§4 

responding  fee,  but  for  each  fee  there  corresponds  not  an  order  of 
a  single  value,  but  orders  of  a  considerable  range  in  value.  This  is 
quite  different  from  the  cases  described  in  Fig.  5.  There  for 
each  reading  Fahrenheit  there  corresponds  a  certain  reading 
centigrade,  and  vice  versa,  and  for  any  change,  however  small,  in  one 
of  the  temperature  readings  a  change,  also  small,  takes  place  in 
the  other  reading.  For  this  reason  the  latter  quantity  is  said  to  be 
continuous. 

The  relation  between  the  temperature  scales  has  been  expressed 
as  an  algebraic  equation.  The  relation  between  the  value  of  a 
money  order  and  the  corresponding  fee  cannot  be  expressed  by  a 
similar  equation.  If  we  had  given  only  a  short  piece  of  the  centi- 
grade-Fahrenheit double  scale,  we  could,  nevertheless,  produce  it 
indefinitely  in  both  directions,  and  hence  find  the  corresponding 
readings  for  all  desired  temperatures.  But  by  knowing  the  fees 
for  a  certain  range  of  money  orders  one  cannot  determine  the  fees 
for  other  amounts.  In  both  of  these  cases,  however,  we  express 
the  fact  of  dependence  of  one  number  upon  another  number  by 
saying  that  the  first  number  is  a  function  of  the  second  number. 

Definition.  Any  number,  u,  is  said  to  be  a  function  of  another 
number,  t,  if,  when  t  is  given,  the  value  of  u  is  determined.  The 
number  t  is  often  called  the  argument  of  the  function  u. 

Illustrations.  The  length  of  a  rod  is  a  function  of  its  tempera- 
ture. The  area  of  a  square  is  a  function  of  the  length  of  a  side. 
The  area  of  a  circle  is  a  function  of  its  radius.  The  square  root 
of  a  number  is  a  function  of  the  number.  The  strength  of  an  iron 
rod  is  a  function  of  its  diameter.  The  pressure  in  the  ocean  is  a 
function  of  the  depth  below  the  surface.  The  price  of  a  railroad 
ticket  is  a  function  of  the  distance  to  be  travelled. 

It  is  obvious  that  any  mathematical  expression  is,  by  the  above 
definition,  a  function  of  the  letter  or  letters  that  occur  in  it. 
Thus,  in  the  equations: 

u  ==  t^  -\-At-\-  1 

t-  1 
^  ~2t-\-2 

u  is  in  each  case  a  function  of  t. 


§4]        VARIABLES  AND  FUNCTIONS  OF  VARIABLES        11 

Temperature  Fahrenheit  is  a  function  of  temperature  centigrade. 
The  value  of  the  fee  paid  for  a  money  order  is  a  function  of  the 
amount  of  the  order. 

Goods  sent  by  freight  are  classified  into  first,  second,  third, 
fourth,  and  fifth  classes.  The  amount  of  freight  on  a  package  is 
a  function  of  its  class.  It  is  also  a  function  of  its  weight.  It  is 
also  a  function  of  the  distance  carried.  Only  the  second  of  these 
functional  relations  just  named  can  readily  be  expressed  by  an 
algebraic  equation.  It  is  possible,  however,  to  express  all  three 
graphically  by  means  of  parallel  scales.  The  definition  of  the  func- 
tion is  given  (for  any  partictdar  railroad)  by  the  complete  freight 
tariff  boo^  of  the  railroad. 

The  fee  charged  for  a  money  order  is  a  function  of  the  amount  of 
the  order.  The  functional  relation  has  been  expressed  graphically 
in  Fig.  6.  Note  that  for  orders  of  certain  amounts,  namely, 
$2§,  $5,  $10,  $20,  $30,  $40,  $50,  $60,  $75,  the  function  is  not  de- 
fined. The  graph  alone  cannot  define  the  function  at  these 
values,  as  one  cannot  know  whether  the  higher,  the  lower,  or  an 
intermediate  fee  should  be  demanded.  One  can,  however,  define 
the  function  for  these  values  by  the  supplementary  statement  (for 
example) :  "For  the  critical  amounts,  always  charge  the  higher  fee." 
As  a  matter  of  fact,  however,  the  lower  fee  is  always  charged. 

A  function  having  sudden  jumps  like  the  one  just  considered,  is 
said  to  be  discontinuous. 

Exercises 

In  the  following  exercises  the  function  described  can  be  represented 
by  a  mathematical  expression.  The 'problem  is  to  set  up  the  expres- 
sion in  each  case. 

1.  One  side  of  a  rectangle  is  10  feet.     Express  the  area  A  as  a 
function  of  the  other  side  x. 

2.  One  leg  of  a  right  triangle  is  15  feet.     Express  the  area  A  as  a 
function  of  the  other  leg  x. 

3.  The  base  of  a  triangle  is  12  feet.     Express  the  area  as  a  func- 
tion of  the  altitude  I. 

4.  Express  the  circumference  of  a  circle  as  a  function  (1)  of  its 
radius  r;  (2)  of  its  diameter  d. 

6.  Express  the  diagonal  d  of  a  square  as  a  function  of  one  side  x. 


12  ELEMENTARY  MATHEMATICAL  ANALYSIS  [§4 

6.  One  leg  of  a  right  triangle  is  10.  Express  the  hypotenuse  h 
as  a  function  of  the  other  leg  x. 

7.  A  ship  B  sails  on  a  course  AB  perpendicular  to  OA.  If  OA  —  30 
miles,  express  the  distance  of  the  ship  from  0  as  a  function  oi  AB. 

8.  A  circle  has  a  radius  10  units.  Express  the  length  of  a  chord 
as  a  function  of  its  distance  from  the  center. 

9.  An  isosceles  triangle  has  two  sides  each  equal  to  15  cm.,  and 
the  third  side  equal  to  x.  Express  the  area  of  the  triangle  as  a 
function  of  x. 

10.  A  right  cone  is  inscribed  in  a  sphere  of  radius  12  inches.  Ex- 
press the  volume  of  the  cone  as  a  function  of  its  altitude  L 

11.  A  right  cone  is  inscribed  in  a  sphere  of  radius  a.  Ej^press  the 
volume  of  the  cone  as  a  function  of  its  altitude  I. 

12.  One  dollar  is  at  compound  interest  for  20  years  at  r  per  cent. 
Express  the  amount  A  as  a  function  of  r. 

Functional  Notation.  The  following  notation  is  used  to  ex- 
press that  one  number  is  a  function  of  another;  thus,  if  w  is  a 
function  of  t  we  write: 

Likewise, 

y  =  /(^) 

means  that  yisa,  function  of  x.     Other  symbols  commonly  used  to 
express  functions  of  x  are: 

0(a;),  X{x),  fix),  Fix),  etc. 

These  may  be  read  the  ''(^-function  of  x,"  the  " X-function of  x," 
etc.,  or  more  briefly,  "the  </>  of  x,''  "the  X  of  .t,"  etc. 

Expressing  the  fact  that  temperature  reading  Fahrenheit  is  a 
function  of  temperature  reading  centigrade,  we  may  write: 

F  =  fiO 

This  is  made  specific  by  writing: 

F  =  tC  +  S2 

Likewise  the  fact  that  the  charge  for  freight  is  a  function  of  class, 
weight,  and  distance,  may  be  written: 

r  =  fie,  w,  d) 


§5]        VARIABLES  AND  FUNCTIONS  OF  VARIABLES        13 

To  make  this  functional  symbol  explicit,  might  require  that  we  be 
furnished  with  the  complete  schedule  as  printed  in  the  freight  tariff 
book  of  the  railroad.  The  dependence  of  the  tariff  upon  class  and 
weight  can  usually  be  readily  expressed,  but  the  dependence  upon 
distance  often  contains  arbitrary  elements  that  cause  it  to  vary 
irregularly,  even  on  different  branches  of  the  same  railroad.  A 
complete  specification  of  the  functional  symbol  /  would  be  con- 
sidered given  in  this  case  when  the  tariff  book  of  the  railroad  was  in 
our  hands. 

5.  Variables  and  Constants.  In  elementary  algebra,  a  letter  is 
always  used  to  stand  for  a  number  that  preserves  the  same  value 
in  the  same  problem  or  discussion.  Such  numbers  are  called 
constants.  In  the  discussion  above  we  have  used  letters  to  stand 
for  numbers  that  are  assumed  not  to  preserve  the  same  value  but 
to  change  in  value;  such  numbers  (and  the  quantities  or  magnitudes 
which  they  measure)  are  called  variables. 

If  r  stands  for  the  distance  of  the  center  of  mass  of  the  earth  from 
the  center  of  mass  of  the  sun,  r  is  a  variable.  In  the  equation 
s  =  hgt^  (the  law  of  falling  bodies),  if  t  be  the  elapsed  time,  s  the 
distance  traversed  from  rest  by  the  falling  body,  and  g  the  acceleration 
due  to  gravity,  then  s  and  t  are  variables  and  g  is  the  constant  32.2 
feet  per  second  per  second. 

The  following  are  constants :  Ratio  of  the  diameter  to  the  circumfer- 
ence in  any  circle;  the  electrical  resistance  of  pure  copper  at  60°  F.; 
the  combining  weight  of  oxygen;  the  density  of  pure  iron;  the  breaking 
strength  of  mild  steel  rods;  the  velocity  of  light  in  empty  space. 

The  following  are  variables:  the  pressure  of  steam  in  the  cylinder  of 
an  engine;  the  price  of  wheat;  the  electromotive  force  in  an  alternating 
current;  the  elevation  of  groundwater  at  a  given  place;  the  discharge 
of  a  river  at  a  given  station.  When  any  of  these  magnitudes  are 
assumed  to  be  measured,  the  numbers  resulting  are  also  variables. 

The  volume  of  the  mercury  in  a  common  thermometer  is  a  variable; 
the  mass  of  mercury  in  the  thermometer  is  a  constant. 

6.  Algebraic  Functions.  An  expression  that  is  built  up  by 
operating  on  x  a  limited  number  of  times  by  addition,  subtraction, 
multiplication,  division,  involution  and  evolution  only,  is  called 
an  algebraic  fimction  of  x.  The  following  are  algebraic  functions 
of  x: 


14  ELEMENTARY  MATHEMATICAL  ANALYSIS  [§6 

(1)  x\  (4)  2x  +  5.         (7)  x3  -  6a;2  +  Ux  -  6. 

x'^  -\-2,x 

(2)  x^  (5)  Ux.  (8)  ^^, 

(3)  3\/^:  (6)  ^2  -  5.  (9)   {x  -  a)  {x  -  b)  {x  -  c). 

The  expression  x^  is  an  algebraic  function  of  x  but  2^  is  not  an 
algebraic  function  of  x.  The  fee  charged  for  a  money  order  is  not 
an  algebraic  function  of  the  amount  of  the  order. 

It  is  convenient  to  divide  algebraic  functions  into  classes.  Thus 
a;2  +  2  is  said  to  be  integral;  (x  +  1)  /(2  —  x^)  and  2  +  x'"^  are 
said  to  be  fractional;  likewise  x^  -\-  2  and  (x  +  1)  /(2  —  a;^)  are 
said  to  be  rational;  \/l  —  x  and  3  —  x^^  are  said  to  be  irrational. 
These  terms  may  be  formally  defined  as  follows: 

An  algebraic  function  of  x  is  said  to  be  rational  if  in  building  up 
the  expression,  the  operation  of  evolution  is  not  performed  upon 
X,  or  upon  a  function  of  x;  otherwise  the  function  is  irrational. 

Thus,  expressions  (1),  (4),  (5),  (6),  (7),  (9),  above,  are  rational 
functions  of  x.  Expressions  (3)  and  (8)  are  irrational.  Ex- 
pression (2)  is  rational  if  n  is  a  whole  number;  otherwise  irrational. 

A  rational  function  is  said  to  be  integral  if  in  building  up  the 
function  the  operation  of  division  by  x,  or  by  a  function  of  x, 
is  not  performed;  otherwise  the  function  is  fractional. 

Thus  expressions  (1),  (4),  (6),  (7),  (9),  above,  are  integral  func- 
tions of  x.  Expressions  (1),  (4),  (6), -(7),  (9)  are  both  rational 
and  integral  and  may  therefore  be  called  rational  integral 
functions  of  x. 

Exercises 

Classify  the  following  functions  of  r,  t,  or  x,  answering  the  following 
questions  for  each  function:  (A)  is  the  function  algebraic  or  (B)  non- 
algebraic?  If  it  is  algebraic,  is  it  (a)  rational  or  (6)  irrational;  if  it 
is  rational,  is  it  (1)  integral  or  (2)  fractional?  The  scheme  of  classifi- 
cation is  as  follows : 


(1)  integral 

(2)  fractional 


A.  Algebraic. 

(a)  rational 

(6)  irrational 

B.  Non-algebraic. 


§7]        VARIABLES  AND  FUNCTIONS  OF  VARIABLES        15 


2.  ax^  +  hx''-  -\-  ex  -\-  d. 

3.  x^  -  xK 

1 

4.  X 


3   +    -   +  X^ 
X 


5.  2x  +  a;^  +  2^  + 


l+<. 


"•  1  -  i       

7.  mx  +  Va2  - 

n.i   —   ^2        /73   — 

8 


(1+0(1  -0;  a  +  V~t){i  -V't). 


3.37x1-86  ^1-25. 

a^  +  x' 


10. 


o  +  x        a  —  X        a  +  x 
9.  (a  -x)(a2  +ax  +  x2);   (c^ 
a  —  ar^ 


1-r 


x^)(a5  +ah^  +x*). 
Write  an  equal  integral  expression. 


7.  Graphical  Computation.     The  ordinary  operations  of  arith- 
metic, such  as  multiplication,  division,  involution  and  evolution, 


Y    U 


<c 

/ 

1 

1 

/ 

/ 
/ 

(A) 

A< 

>C 

A. 

B 

/ 

1 

(BJ 

\l 

> 

AC 
OA 

1 

1 

1 
/ 

B 

1 
1 

/ 

Jl 

j: 

5     6      7     8      9     10 


Fig.  8. — Graphical  Multi- 
plication by  Properties  of 
Similar  Triangles. 


Fig.  9. — Method  of  Graphical  Mul- 
tiplication and  Division  carried  out  on 
Squared  Paper.  The  figure  shows  1.9 
X  4.4  =  8.4. 


can  be  performed  graphically  as  explained  below.  The  graphical 
construction  of  products  and  quotients  is  useful  in  many  problems 
of  science.  The  law  of  proportional  sides  of  similar  triangles  is 
the  fundamental  theorem  in  all  graphical  computation.  Its 
application  is  very  simple,  as  will  appear  from  the  following  work. 


16  ELEMENTARY  MATHEMATICAL  ANALYSIS  [§7 

Problem  1:  To  compute  graphically  the  product  of  two 
numbers.  Let  the  two  numbers  whose  product  is  required  be  a 
and  h.  On  any  Une  lay  off  the  unit  of  measurement,  01,  Fig.  8. 
On  the  same  line,  and,  of  course,  to  the  same  scale,  lay  off  OA 
equal  to  one  of  the  factors  a.  On  any  other  line  passing  through 
1  lay  off  a  line  IB  equal  to  the  other  factor  b.  Join  OB  and 
produce  it  to  meet  AC  drawn  parallel  to  IB.  Then  ^C  is  the 
required  product.     For,  from  similar  triangles: 

AC  :1B  =  OA:  01  (1) 

or, 

AC  =  OA  X  IB  (A) 

It  is  obvious  that  the  angle  OAC  may  be  of  any  magnitude. 
Hence  it  may  conveniently  be  taken  a  right  angle,  in  which  case  the 
work  may  readily  be  carried  out  on  ordinary  squared  paper.  Many 
prefer,  however,  to  do  the  work  on  plain  paper,  laying  off  the 
required  distances  by  means  of  a  boxwood  triangular  scale.  The 
squared  paper,  form  Ml,  prepared  for  use  with  this  book  is  suitable 
for  this  purpose.  On  a  sheet  of  this  paper,  draw  the  two  lines 
OX  and  OF  at  right  angles  and  the  unit  hne  lU,  as  shown  in  Fig. 
9.  Then  from  the  similar  triangles  OIB  and  OAC  the  proportion 
(1)  and  the  formula  (A)  above  are  true.  Hence  to  compute 
graphically  the  product  of  two  numbers  a  and  b  count  off  (Fig.  9) 
OA  =  a  to  the  OX-scale  and  IB  =  b  to  the  OF-scale.  Lay  a 
straight  edge  or  edge  of  a  transparent  triangle  down  to  draw  OC. 
It  is  not  necessary  to  draw  OC,  but  merely  to  locate  the  point  C. 
Then  count  off  AC  to  the  OF-scale.  Then  AC  =  aXbhy  (A). 
The  figure  as  drawn  shows  the  product  4.4  X  1.9  =  8.4. 

All  numbers  can  be  multiplied  graphically  on  a  section  of 
squared  paper  10  units  in  each  dimension  by  properly  reading  the 
OX  and  OY  scales.  Any  product  ab  can  be  written  aibi  X  10'^  = 
Ci  X  10**,  where  ai  and  6i  each  have  one  digit  before  the  decimal 
point,  and  Ci  ^100. 

Thus: 

440  X  19  =  4.40  X  1.9  X  lO^  =  8.40  X  lO^ 
also 

37  X  73  =  3.7    X  7.3  X  10^  =      27  X  10^ 

To  proceed  with  the  product  of  Oi  X  &i,  we  first  determine  by 


§7]        VARIABLES  AND  FUNCTIONS  OF  VARIABLES        17 

inspection  whether  Ci  >  or  <  10.  If  Ci  <  10,  we  read  the  scales 
as  they  are  in  Fig.  9  when  counting  off  ai,  6i  and  Ci.  If  Ci  >  10,  we 
read  the  OX  scale  as  it  stands  when  counting  off  ai  but  read  the 
OY  scale  0,  10,  20,  30,  etc.,  in  counting  off  the  numbers  hi  and  Ci. 

Exercises 

In  using  form  Ml  for  the  following  exercises  take  the  scale  OF  at 
the  left  marginal  line  of  the  sheet  and  use  2  cm.  as  the  unit  of  measure. 
,    Compute  graphically  the  following  products :  Check  results : 


1. 

2.5    X4.8. 

4. 

78.5  X  16.5. 

2. 

4.15  X6.25. 

6. 

2.14  X  0.0467. 

3. 

3.14  X  7.22. 

6. 

2140  X  0.0467. 

Problem  2:     To    compute    graphically    the    quotient    of   two 
numbers  a  and  b.     Formula  (A)  above  can  be  written: 

1B=^  (B) 

From  this  it  is  seen  that  the  quotient  of  two  numbers  a  and  b  can 
readily  be  computed  graphically  by  use  of  Figs.  8  or  9.  In  Fig.  9 
count  off  OA  =  b,  the  divisor,  to  the  OX  scale,  and  AC  =  a,  the 
dividend,  to  the  OF  scale.  Lay  the  triangle  down  to  draw  OC. 
Do  not  draw  OC,  but  mark  the  point  B  and  count  off  IB  to  the 
OY  scale.  Then  IB  =  a/b  by  (B).  Fig.  9  shows  the  quotient 
8.4  -^  4.4  =  1.9.     Any  quotient  a/b  may  be  written 

^  X  10«  =  Q  X  10« 

where  N,  D,  Q  are  each  ^  10  but  >  1.  Hence,  the  OX  and  OY 
scales  may  always  be  read  as  they  stand  in  Fig.  9. 


Exercises 

Compute  graphically  the  following  quotients:  Check  results: 

1.  6.2/2.5.  4.  7.32/1.25. 

2.  1.33/6.45.  5.     872/321. 

3.  234/0.52.  6.     128/937. 


18 


ELEMENTARY  MATHEMATICAL  ANALYSIS 


[§7 


Problem  3:  To  compute  graphically  the  square  root  of  any 
number  N.  In  Fig,  10  count  ofl  lA  =  N  to  the  OX  scale,  and 
draw  a  semicircle  on  OA  as  a  diameter.  Then  IC  =  \/N  to 
the  OY  scale.  Another  construction  is  to  place  the  triangle  in 
the  position  shown  in  Fig.  10,  so  that  the  two  edges  pass  through 
0  and  A  and  the  vertex  of  the  right  angle  lies  on  the  line  lU. 
Fig.  10  shows  the  construction  for  \/7.     The  readings  on  the  OX 


10^!^ 

9 

8 

n 

f 

y 

s 

n 

H 

s 

II 

--- 

^^ 

\ 

n 

1 

B 

^^ 

A 

X 

L    \ 

\  I 

!    ^ 

\ 

i 

?  i 

""~ 

)-l 

Q 

' 

-^ 

"^ 

^^ 

/ 

/ 

"■-~ 

~^. 

" 

-- 

1 

1 

( 

— 

-- 

-" 

/ 

1 

Fig.  10. — Graphical  Method  of  the  Extraction  of    Square    Roots.     The 
figure  shows  •\/7  =  2.65. 

scale  may  be  multipHed  by  lO^'*  and  those  on  the  OY  scale  by  10" 
where  n  is  any  integer  positive  or  negative. 

State  the  two  theorems  in  plane  geometry  on  which  the  proof  of 
these  two  constructions  depends. 

Problem  4:  To  compute  graphically  the  square  of  any  num- 
ber N.     This  is  a  special  case  of  Problem  1,  when  a  =  b  =  N. 


Exercises 

1.  Compute  the  square  roots  of  2,  3,  5,  and  7. 

2.  Compute  the  square  roots  of  3.75,  37.5,  0.375. 

3.  Compute  the  squares  of  1.23  and  3.45. 

4.  Compute  the  squares  of  7.75  and  0.895. 

5.  Show  that  w^  is  nearly  10. 


§7]        VARIABLES  AND  FUNCTIONS  OF  VARIABLES        19 

Problem  5:  To  compute  graphically  the  reciprocal  of  any 
number  N.  This  is  a  special  case  of  Problem  2,  when  a  =  1  and 
b  =  N. 

Problem  6:  To  compute  graphically  the  integral  powers  of 
any  number  N.  This  problem  is  solved  by  the  successive  applica- 
tion of  Problem  1  to  construct  iV^,  N^,  N*,  etc.,  and  of  Problem  2 


Ui 


5 

/ 

/ 

/ 

/ 

4 

1 

/ 

/ 

/ 

1 

"7 

3 

1 

/ 

1 

/ 

/ 

2 

'/ 

1 

1 

1/ 

/ 

/// 

// 

.1 

% 

^ 

^ 

-3 
-4 

0  \      N     Z  3  4 

Fig.   11. — Graphical  Computation  of    (1.5)'*  for  n  =   -4,  —3,    —2,-1 
0,  1,  2,  3,  4,  5. 


to  construct  iV-^  -^"^j  ^~^  ^tc.     This  construction  is  shown  for 
the  powers  of  1.5  in  Fig.  11. 

Exercises 

1.  Compute  the  reciprocal  of  2.5;  of  3.33;  of  0.75;  of  7.5. 

2.  Compute  (1.2)',  (0.85)\  (1.15)*. 


20 


ELEMENTARY  MATHEMATICAL  ANALYSIS 


[§8 


3.  Show  that  (1.05)^^  =  2.08,  so  that  money  at  5  percent  com- 
pound interest  more  than  doubles  itself  in  fifteen  years. 

Note:  The  work  is  less  if  (1.05)^  is  first  found  and  then  this  result 
cubed. 

4.  From  the  following  outline  the  student  is  to  produce  a  complete 
method,  including  proof,  of  constructing  successive  powers  of  any 
number. 

Let  OA  (Fig.  12)  be  a  radius  of  a  circle  whose  center  is  0.  Let 
OB  be  any  other  radius  making  an  acute  angle  with  OA.  From  B 
drop  a  perpendicular  upon  OA,  meeting  the  latter  at  Ai.  From  Ai 
drop  a  perpendicular  upon  OB  meeting  OB  at  A 2.  From  A 2  drop  a 
perpendicular  upon  OA  meeting  OA  at  A3,  and  so  on  indefinitely. 
Then,  if  OA  be  unity,  OAj  is  less  than  unity,  and  OA2,  OAs,  OAi 
.    .    .  are,  respectively,  the  square,  cube,  fourth  power,  etc.,  of  OAi. 


Fig.   12. — Graphical    Computation   of   Powers    of   a    Number. 

Instead  of  the  above  construction,  erect  a  perpendicular  to  OB  meet- 
ing OA  produced  at  ai.  At  ai  erect  a  perpendicular  meeting  OB  pro- 
duced at  a2,  and  so  on  indefinitely.  Then  if  OA  be  unity,  ai  is 
greater  than  unity  and  a^,  az,  ai,  .  .  .  are,  respectively,  the  square, 
cube,  etc.,  of  ai.     As  an  exercise,  construct  powers  of  4/5  and  of  2.5. 

5.  Show  that  the  successive  "treads  and  risers"  of  the  steps  of 
the  "stairways"  of  Figs.  13  and  14  are  proportional  to  the  powers 
of  r.  The  figures  are  from  Milaukovitch,  Zeitschrift  fiir  Math, 
und  Nat.  Unterricht,  Vol.  40,  p.  329. 

8.  Double  Scales  for  Several  Simple  Algebraic  Functions.     We 

may  make  use  of  the  graphical  method  of  computation  explained 


§8]        VARIABLES  AND  FUNCTIONS  OF  VARIABLES        21 

above  to  construct  graphically  double  scales  representing  simple 
algebraic  relations.  For  example,  we  may  construct  a  double 
scale   for    determining    the    square    of    anj'-    desired    number. 


E 

/        ar^ 

/   ar'^ 

/ 

C 

/.r- 

/ 

y 

^D 

/ox 

^v^^^ 

/a    \ 

/ 

B  M        A  B 

Fig.   13.  Fig.   14. 

Computation  of  ar,  ar^,  ar^,  .  .  .  for  r  <  1  and  for  r  >  1. 

Call  OA  (see  Fig.  15)  the  scale  on  which  we  desire  to  read  the 
number;  call  OB  the  scale  on  which  we  read  the  square.  Let 
us  agree  to  lay  off  OA  as  a  uniform  scale,  using  01  as  the  unit  of 
measure.     Since   we  desire  to    read  opposite  0,  1,  2,  3,  of  the 


u  ^                       ^"^-^^^ 

A 

B    A 

^-^'^            ^^"^^""^^     ^\     \ 

3- 

^^^^""■"^^-T^^        \    \ 

2^ 

■e: 

— ^^^^\  \\  \  \ 

^ 

■^r"^^\  \  \  \  \  \  \ 

1- 

-i. 

N.                  \             \             \              \              \              \              \ 

0: 

^ 

\    \    \    1    1    1    1    1 

Fig.  15.— Method  of  Constructing  a  Double  Scale  of  Squares  or  of  Square 

Roots. 


uniform  scale,  the  squares  of  these  numbers,  the  lengths  along  the 
scale  OB  must  be  laid  off  proportional  to  the  square  roots  of  the  num- 
bers 0,  1,  2,  3,   .    .   ,  that  is,  the  square  root  of  any  length,  when 


22 


ELEMENTARY  MATHEMATICAL  ANALYSIS 


:§9 


laid  off  on  OB,  and  marked  with  the  symbol  of  the  original  length, 
will  he  opposite  the  square  root  of  that  number  on  OA. 

No  difficulty  need  be  experienced  in  carrying  out  the  actual 
construction  of  double  scales  representing  algebraic  relations, 
either  by  use  of  a  table  of  numerical  values  of  the  function  or  by 
means  of  graphical  construction.  As  a  less  laborious  method  of 
graphically  expressing  functional  relations  will  be  explained  in  the 
next  chapter,  the  matter  of  double  scales  will  not  be  discussed 
further  at  this  place. 

'N 


Fig.  16.- 


-The  Fahrenheit-centigrade  Double  Scale  Opened  about  the 
32°  Mark  of  the  Fahrenheit  Scale  as  Pivot. 


9.  Fimctions  Represented  by  Scales  not  in  Juxtaposition.     It  is 

obvious  that  any  double  scale  used  to  express  the  relation  between 
a  function  of  a  variable  and  the  variable  itself,  may  be  separated,  if 
desired,  into  two  distinct  scales,  provided  means  be  adopted  for 
connecting  corresponding  points  on  the  two  scales.  For  example, 
one  of  the  two  scales  may  be  rotated  about  any  one  of  its  points, 
as  scissors  about  their  pivot,  thereby  forming  two  intersecting 
straight  lines.  Corresponding  points  may  then  be  connected  by 
erecting  perpendiculars   to   each   scale   and   joining   those   that 


§9]       VARIABLES  AND  FUNCTIONS  OF  VARIABLES 


23 


proceed  from  corresponding  points,  or  by  any  other  practical  means. 
In  Fig.  16  the  centigrade  and  Fahrenheit  scales  are  shown  opened 
about  the  32°  division  of  the  Fahrenheit  scale  as  pivot.  Perpen- 
diculars erected  at  corresponding  points  of  the  two  scales  meet  at 
the  points  Pi,  P^,  Pz,   .    .    . 

The  line  NOM  on  which  these  points  lie  is  straight.  Why? 
The  student  will  write  out  a  proof,  making  use  of  any  three  points 
as  Pi,  Pi,  Pz,  and  a  property  of  similar  triangles.  Of  course  the 
angle  between  OC  and  OF  need  not  be  taken  as  a  right  angle. 


Fig.  17.- 


-Same  as  Fig.  16  with  the  Lengths  of  the  Units  on  the  OC  and  OF 
Scales  Made  the  Same. 


It  is  also  obvious  that  the  divisions  on  both  scales  may  now  be 
made  the  same  length;  that  is,  OQx,  OQ2,  OQi,  .  .  .  may  be 
made  the  same  length  as  ORi,  OR2,  OR3,  ....  This  is  at 
once  accomplished  if  the  lines  OQi,  OQ2,  OQ3,  .  .  .  ,  be  each 
elongated  in  the  ratio  of  ORi/OQi.  The  functional  relation  may 
be  expressed  equally  well  by  marking  as  before  the  intersection 


24  ELEMENTARY  MATHEMATICAL  ANALYSIS  [§9 

of    the   perpendiculars    erected    at  corresponding  values.    The 
result  is  shown  in  Fig.  17. 

In  the  same  manner  any  of  the  double  scales  may  be  opened 
about  any  point  as  pivot.  If  the  angle  between  the  scales  is 
made  90°,  the  relation  between  the  function  and  its  argument  is 
shown  by  points  on  a  straight  Une  making  an  angle  of  45°  with  each 
scale.  If  one  of  the  scales  be  non-uniform,  it  may,  after  it  is 
turned  about  the  selected  pivot,  be  made  a  uniform  scale,  in  which 
case  the  straight  line  just  mentioned  becomes,  in  general,  a  curved 
line.  We  see,  therefore,  that  instead  of  showing  the  relation 
between  a  function  and  its  variable  by  means  of  two  scales  in 
juxtaposition,  we  may  use  two  uniform  scales  intersecting  at  an 
angle,  and  connect  corresponding  values  of  the  variable  and  its 
function  by  perpendiculars  erected  at  these  corresponding  points. 
The  pairs  of  perpendiculars  intersect  at  points  which,  in  general, 
lie  upon  a  curve.  This  curve  is  obviously  characteristic  of  the 
particular  functional' relation  under  discussion.  The  respresenta- 
tion  of  functional  relations  in  this  manner  leads  to  the  considera- 
tion of  so-called  coordinate  systems,  the  discussion  of  which  is 
begun  in  the  next  chapter. 


CHAPTER  II 


RECTANGULAR    COORDINATES   AND    THE    POWER 
FUNCTION 

10.  Rectangular  Coordinates.  Two  intersecting  algebraic 
with  their  zero  points  in  common,  may  be  used  as  a  system 
of  latitude  and  longitude  to  locate  any  point  in  their  plane.  The 
student  should  be  familiar  with  the  rudiments  of  this  method  from 
the  graphical  work  of  elementary  algebra.     The  scheme  is  illus- 


Y 

4 

s 

p 

2} 

^,Z 

«) 

Pi 

( 

3. 

I) 

? 

1 

X' 

D 

X 

_  * 

-2 

-1 

0 

] 

2 

3 

4 

5 

-1 

P^ 

(- 

2.- 

X) 

-?i 

-8 

yi 

P* 

_r 

2A 

0 

Fig.   18. — Rectangular  Coordinates. 

trated  in  its  simplest  form  in  Fig.  18,  where  one  of  the  horizontal 
lines  of  a  sheet  of  squared  paper  has  been  selected  as  one  of  the 
algebraic  scales  and  one  of  the  vertical  lines  of  the  squared  paper 
has  been  selected  for  the  second  algebraic  scale.  To  locate  a  given 
point  in  the  plane  it  is  merely  necessary  to  give,  in  a  suitable  unit 
of  measure  (as  centimeter,  inch,  etc.),  the  distance  of  the  point  to 
the  right  or  left  of  the  vertical  scale  and  its  distance  above  or  below 

25 


26  ELEMENTARY  MATHEMATICAL  ANALYSIS         [§10 

the  horizontal  scale.  Thus  the  point  P,  in  Fig.  18,  is  2|  units  to 
the  right  and  3|  units  above  the  standard  scales.  P2  is  3  units  to 
the  left  and  2  units  above  the  standard  scales,  etc.  Of  course 
these  directions  are  to  be  given  in  mathematics  by  the  use  of  the 
signs  ''-{-"  and  ''  —  "  of  the  algebraic  scales,  and  not  by  the  use 
of  the  words  ''right"  or  ''left,"  "up"  or  "down."  The  above 
scheme  corresponds  to  the  location  of  a  place  on  the  earth's 
surface  by  giving  its  angular  distance  in  degrees  of  longitude  east 
or  west  of  the  standard  meridian,  and  also  by  giving  its  angular 
distance  in  degrees  of  latitude  north  or  south  of  the  equator. 

The  sort  of  latitude  and  longitude  that  is  set  up  in  the  manner 
described  above  is  known  in  mathematics  as  a  system  of  rectangu- 
lar coordinates.  It  has  become  customary  to  letter  one  of  the 
scales  XX',  called  the  X-axis,  and  to  letter  the  other  YY',  called 
the  Y-axis.  In  the  standard  case  these  are  drawn  to  the  right 
and  left,  and  up  and  down,  respectively,  as  shown  in  Fig.  18. 
The  distance  of  any  point  from  the  F-axis,  measured  parallel  to 
the  X-axis,  is  called  the  abscissa  of  the  point.  The  distance  of 
any  point  from  the  X-axis,  measured  parallel  to  the  F-axis,  is 
called  the  ordinate  of  the  point.  Collectively,  the  abscissa  and 
ordinate  are  spoken  of  as  the  coordinates  of  the  point.  Abscissa 
corresponds  to  the  longitude  and  ordinate  corresponds  to  the 
latitude  of  the  point,  referred  to  the  X-axis  as  equator,  and  to 
the  F-axis  as  standard  meridian.  In  the  standard  case,  abscissas 
measured  to  the  right  of  YY'  are  reckoned  positive,  those  to  the 
left,  negative.  Ordinates  measured  up  are  reckoned  positive, 
those  measured  down,  negative. 

Rectangular  coordinates  are  frequently  called  Cartesian  co- 
brdinates,  because  they  were  first  introduced  into  mathematics 
by  Ren6  Descartes  (1596-1650). 

The  point  of  intersection  of  the  axes  is  lettered  0  and  is  called 
the  origin.  The  four  quadrants,  XOY,  YOX',  X'OY',  Y'OX,  are 
called  the  first,  second,  third,  and  fourth  quadrants,  respectively. 

A  point  is  designated  by  writing  its  abscissa  and  ordinate  in  a 
parenthesis  and  in  this  order:  Thus,  (3,  4)  means  the  point  whose 
abscissa  is  3  and  whose  ordinate  is  4.  Likewise  (  —  3,  4)  means  the 
point  whose  abscissa  is  (—  3)  and  whose  ordinate  is  (+  4). 

Unless  the  contrary  is  explicitly  stated,  the  scales  of  the  co- 


§11]  RECTANGULAR  COORDINATES  27 

ordinate  axes  are  assumed  to  be  straight  and  uniform  and  to  inter- 
sect at  right  angles.  Exceptions  to  this  are  not  uncommon, 
however,  of  which  examples  are  given  in  Figs.  19  and  22. 

The  use  of  two  intersecting  algebraic  scales  to  locate  individual 
points  in  the  plane,  as  explained  above,  is  capable  of  immediate 
enlargement.  It  will  be  explained  below  that  a  suitable  array,  or 
set,  or  locus  of  such  points  may  be  used  to  exhibit  the  relation 
between  two  variables  laid  off  on  the  two  scales,  or  between  a 
variable  laid  off  on  one  of  the  scales  and  a  function  of  the  variable 
laid  off  on  the  other  scale.  This  fact  has  already  been  explained 
from  another  point  of  view  at  the  close  of  the  preceding  chapter. 

11.  Statistical  Graphs.  From  work  in  elementary  algebra  the 
student  is  supposed  to  be  famihar  with  the  construction  of  statis- 


s 
-^ 

s 
^ 

s" 

S* 
(^ 

s 

a 

?3 

i-i 

(M 

« 

'J" 

600 

500 

Fig.  19. — Barograph  Taken  During  a  Balloon  Journey.     The  vertical 
scale  is  atmospheric  pressure  in  millimeters  of  mercury. 

tical  graphs  similar  to  those  presented  in  Figs.  19  to  32.  The 
student  will  study  each  of  these  graphs  and  the  following  brief 
descriptions  before  making  any  of  the  drawings  required  in  the 
exercises  that  follow. 

Fig.  19  is  a  barograph,  or  autographic  record  of  the  atmospheric 
pressure  recorded  November  24,  1907,  during  a  balloon  journey 
from  Frankfort  to  Marienburg  in  West  Prussia.  One  set  of  scales 
consists  of  equal  circles,  the  other  of  parallel  straight  lines.  The 
zero  of  the  scale  of  pressure  does  not  appear  in  the  diagram. 
Note  also  that  the  scale  of  pressure  is  an  inverted  scale,  increasing 
downward.  The  scale  of  time  is  an  algebraic  scale,  the  zero  of 
which  may  be  arbitrarily  selected  at  any  convenient  point.  The 
scale  of  pressure  is  an  arithmetical  scale.  The  zero  of  the  baro- 
metric scale  corresponds  to  a  perfect  vacuum — no  less  pressure 
exists. 


28  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§11 


Chicago 


A.M. 
0       2       4       6 


^^oon  ^-  M 

10      12       2       4        0        8       10      12 


"Woodstcok 

Harvard 

62.'! 
Janesvllle 

91.1 

Madison 
129.7 

Baraboo 

172.6 
Elroy 

203.9 


Eau  Claire 

322.1 
INIenomonie 

Jet. 

3352 
Hudson 


St.  Paul 

398.0    /o       2       4 

Minneap-A.  M. 
oils 

408.4 


/ 

/ 

/' 

V 

\ 

/ 

\ 

/^ 

/ 

A 

) 

\ 

\ 

\ 

/ 

/ 

1 

'\ 

J 

% 

\ 

4jl 

'f 

/ 

/ 

\ 

/ 

/ 

\ 

\ 

— N; 

ij 

/ 

/ 

\ 

/ 

\ 

A 

/ 

V 

\ 

\ 

\ 

\ 

/ 

/ 

/^ 

\ 

\ 

/ 
/  ( 

/ 

V 

\ 

\ 

i/ 

¥ 

/ 

\ 

\ 

\ 

/ 

V 

^ 

^ 

-^ 

X 

^ 

^ 

s-f 

8       10      12       2       4 
Noon 


10      12 
P.M. 


Fig.  20. — Graphical  Time-table  of  Certain  Railway  Trains  between 
Chicago  and  Minneapolis. 


Monday  Tuesday       Wednesday      Thursday  Friday  Saturday 


Fig.  21. — Graphical  Time-table  of  Passenger  Trains  between  Chicago 
and  Los  Angeles. 


§11] 


RECTANGULAR  COORDINATES 


29 


Fig.  20  is  a  graphical  time-table  of  certain  passenger  trains  be- 
tween Chicago  and  Minneapolis.  The  curves  are  not  continuous, 
as  in  the  case  of  the  barograph,  but  contain  certain  sudden  jumps. 
What  is  the  meaning  of  these?     What  indicates  the  speed  of  the 


Fig.  22. — Upper  Curve,  Elevation  of  Water  in  a  Well  on  Long  Island 
Lower  curve,  elevation  of  water  in  the  nearby  ocean. 


trains?    Where  is  the  fastest  track  on  this  railroad?     What  shows 
the  meeting  point  of  trains? 

//  the  diagram,  Fig.  20,  be  wrapped  around  a  vertical  cylinder  of 
such  size  that  the  two  midnight  lines  just  coincide,  then  each  train  line 
may  he  traced  through  continuously  from  terminus  to  terminus. 


30 


ELEMENTARY  MATHEMATICAL  ANALYSIS         [§11 


Functions  having  this  remarkable  property  are  said  to  be  periodic. 
In  the  present  case  the  trains  run  at  the  same  time  every  day, 
that  is,  periodically.  In  mathematical  language,  the  position  of 
the  trains  is  said  to  be  a  periodic  function  of  the  time. 

Fig.  21  is  the  graphical  time-table  of  'limited"  trains  between 
Chicago  and  Los  Angeles.  The  schedule  of  train  No.  1,  a  very 
heavy  passenger  train,  is  placed  upon  the  chart  for  comparison. 
The  periodic  character  of  this  function  is  brought  out  very  clearly 
by  using  time  as  the  abscissa.     The  student  should  discuss  the 


t:-60 


-50 


bo 


-40 


^10 


iliiii! 


10         20         30         40         50         60         70*       80         90        100 
Amount  of  the  Money  Order  in  Dollars 

Fig.  23. — The  Graph  of  a  Discontinuous  Function. 

discontinuities  and  the  various  speeds  as  shown  from  the  diagram. 
The  track  profile  is  given  at  the  right  of  the  diagram  for  purposes 
of  comparison. 

Fig.  22  represents  the  fluctuation  of  the  elevation  of  the  ground- 
water at  a  certain  point  near  the  sea-coast  on  Long  Island.  The 
fluctuations  are  primarily  due  to  the  tidal  wave  in  the  near-by 
ocean.  Here  the  scale  of  one  of  the  coordinates  (elevation)  is 
laid  off  on  a  series  of  equal  circumferences  similar  to  those  of  Fig. 
19.  The  scale  of  the  other  coordinate  (time)  is  laid  off  on  the  mar- 
gin of  the  outer  or  bounding  circle.  The  curve  is  continuous. 
Is  the  curve  periodic?  What  indicates  the  rate  of  change  in  the 
elevation  of  the  ground-water?  When  is  the  elevation  changing 
most  rapidly?     When  is  it  changing  most  slowly? 


§12]  RECTANGULAR  COORDINATES  31 

Fig.  23  represents  the  functional  relation  between  the  amount  of 
a  domestic  money  order  and  the  fee.  Two  arithmetical  scales 
were  used  in  making  the  diagram,  as  in  ordinary  rectangular  co- 
ordinates, except  that  the  vertical  scale  is  ten-fold  the  horizontal 
scale;  that  is,  lengths  that  represent  dollars  on  the  one  scale  rep- 
resent cents  on  the  other.  This  is  an  excellent  illustration  of  a 
discontinuous  function.  On  account  of  the  sudden  jumps  in  the 
values  of  the  fee,  the  fee,  as  explained  in  the  preceding  chapter,  is 
said  to  be  a  discontinuous  function  of  the  amount  of  the  order. 

12.  Suggestions  on  the  Construction  of  Graphs.  Two  kinds  of 
rectangular  coordinate  paper  have  been  prepared  for  use  with  this 
book.  Form  Ml  is  ruled  in  centimeters  and  fifths,  and  permits 
two  scales  of  twenty  and  twenty-five  major  units  respectively  to 
be  laid  off  horizontally  and  vertically  on  a  standard  sheet  of  letter 
paper  8|  X  11  inches.  Form  M2  is  ruled  without  major  divisions 
in  uniform  1  /5-inch  intervals.  This  form  of  ruling  is  desirable  for 
general  computation  and  for  graphing  functions  for  which  non- 
decimal  fractional  intervals  are  used,  such  as  eighths,  twelfths, 
or  sixteenths,  which  often  occur  in  the  measurement  of  mass 
or  time. 

It  is  a  mistake  to  assume  that  more  accurate  work  can  be  done 
on  finely  ruled  than  on  more  coarsely  ruled  squared  paper.  Quite 
the  contrary  is  the  case.  Paper  ruled  to  1  /20-inch  intervals  does 
not  permit  interpolation  within  the  small  intervals  while  paper 
ruled  to  1  /lO  or  1  /5-inch  intervals  permits  accurate  interpolation 
to  one-tenth  of  the  smallest  interval.  Form  Ml  is  ruled  to 
2-mm.  intervals,  and  is  fine  enough  for  any  work.  The  centi- 
meter unit  has  the  very  considerable  advantage  of  permitting 
twenty  of  the  units  within  the  width  of  an  ordinary  sheet  of  letter 
paper  (8^  X  11  inches)  while  seven  is  the  largest  number  of  inch 
units  available  on  such  paper. 

In  order  to  secure  satisfactory  results,  the  student  must  recognize 
that  there  are  several  varieties  of  statistical  graphs,  and  that 
each  sort  requires  appropriate  treatment. 

1.  It  is  possible  to  make  a  useful  graph  when  only  one  variable 
is  given.  Thus  the  following  table  gives  the  ultimate  tensile 
strength  of  various  materials: 


32  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§12 

ULTIMATE  TENSILE  STRENGTH  OF  VARIOUS  MATERIALS 


Material 


Tensile  strength, 
tons  per  square  inch 


Hard  steel |  50. 0 

Structural  steel I  30. 0 

Wrought  iron 25 . 0 

Drawn  brass i  21.5 

Drawn  copper I  16 . 0 

Cast  brass I  12.0 

Cast  copper j  11.0 

Cast  iron [  10.0 

Timber,  with  grain [  5.0 


A  graph  showing  these  results  is  given  in  Fig.  24.  There  are 
two  practical  ways  of  showing  the  numerical  values  pertaining 
to  each  material,  both  of  which  are  indicated  in  the  diagram;  either 
rectangles  of  appropriate  height  may  be  erected  opposite  the 
name  of  each  material,  or  points  marked  by  circles,  dots  or  crosses 
may  be  located  at  the  appropriate  height.  It  is  obvious  in  this 
case  that  a  smooth  curve  should  not  be  drawn  through  these  points 
— such  a  curve  would  be  quite  meaningless.  In  this  case  there 
are  not  two  scales,  but  merely  the  single  vertical  scale.  The  hori- 
zontal axis  bears  merely  the  names  of  the  different  materials 
and  has  no  numerical  or  quantitative  significance.  The  result 
is  obviously  not  the  graph  of  a  function,  for  there  are  not  two 
variables,  but  only  one.  The  graph  is  merely  a  convenient  ex- 
pression for  certain  discrete  and  independent  results  arranged 
in  order  of  descending  magnitude. 

2.  It  is  possible  to  have  a  graph  involving  two  variables  in 
which  it  is  either  impossible  or  undesirable  to  represent  the  graph 
by  a  continuous  curve  or  line.  For  example.  Fig.  25  is  a  graph 
representing  the  maximum  temperature  on  each  day  of  a  certain 
month.  Because  there  is  only  one  maximum  temperature  on 
each  day,  the  value  corresponding  to  this  should  be  shown  by  an 
appropriate  rectangle,  or  by  marking  a  point  by  a  circle,  or  by  a 
dot  or  cross,  as  in  the  preceding  case,  since  a  continuous  curve 
through  these  points  has  no  meaning.     The  horizontal  scale  may 


!l2] 


RECTANGULAR  COORDINATES 


33 


be  marked  by  the  names  of  the  days  of  the  week  or  by  numbers, 
but  in  either  case  the  horizontal  line  is  a  true  scale,  as  it  corresponds 
to  the  lapse  of  the  variable  time.  Sometimes,  as  in  Fig.  25, 
graphs  of  this  kind  are  represented  by  marking  the  appropriate 
points  by  dots  or  circles  and  then  connecting  the  successive  points 
by  straight  lines.  These  lines  have  no  special  meaning  in  such 
a  case,  but  they  aid  the  eye  in 
following  the  succession  of  sepa- 
rate points. 

If  a  graph  be  made  of  the  noon- 
day temperatures  of  each  day  of 
the  same  month  referred  to  in 
Fig.  25,  one  of  the  same  methods 
indicated  above  would  be  used 
to  represent  the  results;  that  is, 
either  rectangles,  marked  points, 
or  marked  points  joined  by 
lines.  Although  a  smooth  curve 
drawn  through  the  known  points 
would  have  a  meaning  (if  cor- 
rect), it  is  obvious  that  the  noon- 
day temperatures  alone  are  not 
sufficient  for  determining  its 
form.  In  all  such  cases  a  smooth 
curve  should  not  be  drawn. 

Fig.  26  shows  the  monthly 
output  and  gross  earnings  of  a 
power  company  during  its  first 
months  of  operation;  the  fixed 
charges  are  also  shown  upon  the  same  diagram.  (See  also  Figs. 
24  and  84.) 

3.  If  the  data  are  reasonably  sufficient,  a  smooth  curve  may, 
and  often  should,  be  drawn  through  the  known  points.  Thus  if 
the  temperature  be  observed  every  hour  of  the  day  and  the  results 
be  plotted,  a  smooth  curve  drawn  carefully  through  the  known 
points  will  probably  very  accurately  represent  the  unknown 
temperatures  at  intermediate  times.  The  same  may  safely  be 
done  in  exercises  (1)  and  (2)  below.     In  scientific  work  it  is  desir- 

3 


X 

«;oX 

OU    HK 

fl          1 

a          I 

M                    \ 

^  Al\ 

o                t 

0.                      ^ 

s       i 

S^(\        -,  >- 

H  oU          -*(- 

-d                                ~  t- 

«OA                                        "^S 

£  20  -                  ^ 

Sin                                l"^--'^^ 

g  10                                                 "t" 

^     '                           ^ 

.  1  M  M  1 1 1 

W      00 


2  I   I  I   a 

Q     O     O     O     H 


Fig.  24. — Graph  Showing  Ten- 
sile Strength  of  Certain  Structural 
Materials. 


34 


ELEMENTARY  MATHEMATICAL  ANALYSIS 


(12 


Fig.   25. — Maximum  Daily  Temperatures,  Madison,  Wis.,  February,  1914 


Fig.  26. 


-Graph  of  Monthly  Gross  Earnings  and  Output  of  a  Power  Plant 
During  Initial  Stages  of  Operation. 


§12]  RECTANGULAR  COORDINATES,  35 

able  to  mark  by  circles  or  dots  the  values  that  are  actually  given 
to  distinguish  them  from  the  intermediate  values  "guessed"  and 
represented  by  the  smooth  curve. 

In  addition  to  the  above  suggestions,  the  student  should  adhere 
to  the  following  instructions : 

4.  Every  graph  should  be  marked  with  suitable  numerals  along 
both  numerical  scales. 

5.  Each  scale  of  a  statistical  graph  should  bear  in  words  a 
description  of  the  magnitude  represented  and  the  name  of  the  unit 
of  measure  used.  These  words  should  be  printed  in  drafting  let- 
ters and  not  written  in  script. 

6.  Each  graph  should  bear  a  suitable  title  telling  exactly  what  is 
represented  by  the  graph. 

7.  The  selection  of  the  units  for  the  scale  of  abscissas  and  ordi- 
nates  is  an  important  practical  matter  in  which  common  sense  must 
control.  It  is  obvious  that  in  the  first  exercise  given  below  1  cm. 
=  1  foot  draft  for  the  horizontal  scale,  and  1  cm.  =  100  tons  for 
the  vertical  scale  will  be  units  suitable  for  use  on  form  Ml. 

Further  instruction  in  practical  graphing  is  given  in  §33. 

Exercises 

1.  At  the  following  drafts  a  ship  has  the  displacements  stated: 


Draft  in  feet,  A |       15       |       12       |        9        |     6.3 


Displacement  in  tons,  T i     2096  1512     |     1018 


586 


Plot  on  squared  paper.     What  are  the  displacements  when  the 
drafts  are  11  and  13  feet,  respectively? 

2.  The  following  tests  were  made  upon  a  steam  turbine  generator^ 


Output  in  kilowatts,  K.: 1,190        9951       745]       498|     247 


Weight,  pounds  of  steam  con-  |  23,120:  20,040j  16,630,  12,560 
sumed  per  hour,  W.  'I 


8.320 


Plot  on  squared  paper.     What  are  the  probable  values  of  K  when 
W  is  22,00  0  and  also  when  W  is  11,000? 


36  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§13 

3.  Make  a  graphical  chart  of  the  zone  rates  of  the  Parcel  Post 
Service  for  the  first  three  zones,  using  weight  of  package  as  abscissa 
and  cost  of  postage  as  ordinate. 

4.  The  average  temperature  at  Madison  from  records  taken  at 
7  a.  m.  daily  for  30  years  is  as  follows: 

Jan.    1,  14.0.  F.  July    1,  67.5.  F. 

Feb.  1,  15.1.  Aug.  1,  64.0. 

Mar.  1,35.2.  Sept.  1,  55.4. 

Apr.   1,  40.0.  Oct.    1,  44.1. 

May  1,  53.9.  Nov.  1,  30.0. 

June  1,  63.2.  Dec.    1,  18.3. 

Make  a  suitable  graph  of  these  results  on  squared  paper. 

13.  Mathematical,  or  Non-statistical  Graphs.  Instead  of  the 
expressions  ''abscissa  of  a  point/'  or  ''ordinate  of  a  point/'  it  has 
become  usual  to  speak  merely  of  the  "x  of  a  point,"  or  of  the  "y  of 
a  point,"  since  these  distances  are  conventionally  represented  by 
the  letters  x  and  y,  respectively.  If  we  impose  certain  conditions 
upon  X  and  y,  then  it  will  be  found  that  we  have,  by  that  very  fact, 
restricted  the  possible  points  of  the  plane  located  by  them  to  a 
certain  array,  or  set,  or  locus  of  points,  and  that  all  other  points 
of  the  plane  fail  to  satisfy  the  conditions  or  restrictions  imposed. 

It  is  obvious  that  the  command,  "Find  the  place  whose  latitude 
equals  its  longitude,"  does  not  restrict  or  confine  a  person  to  a  par- 
ticular place  or  point.  The  places  satisfjdng  this  condition  are 
unlimited  in  number.  We  indicate  all  such  points  by  drawing 
a  line  bisecting  the  angles  of  the  first  and  third  quadrants;  at  all 
points  on  this  line  latitude  equals  longitude.  We  speak  of  this 
line  as  the  locus  of  all  points  satisfying  the  conditions.  We  might 
describe  the  same  locus  by  saying  "the  y  of  each  point  of  the 
locus  equals  the  a;,"  or,  with  the  maximum  brevity,  simply  write 
the  equation  "y  =  x."  This  is  said  to  be  the  equation  of  the 
locus,  and  the  line  is  called  the  locus  of  the  equation. 

It  is  of  the  utmost  importance  to  be  able  readily  to  interpret  any 
condition  imposed  upon,  or,  what  is  the  same  thing,  any  relation 
between  variables,  when  these  are  given  in  words.  It  will  greatly 
aid  the  beginner  in  mastering  the  concept  of  what  is  meant  by  the 
term  function  if  he  will  try  to  think  of  the  meaning  in  words  of  the 


§13]  RECTANGULAR  COORDINATES  37 

relations  commonly  given  by  equations,  and  vice  versa.  The 
very  elegance  and  brevity  of  the  mathematical  expression  of  rela- 
tions by  means  of  equations,  tends  to  make  work  with  them  formal 
and  mechanical  unless  care  is  taken  by  the  beginner  to  express  in 
words  the  ideas  and  relations  so  briefly  expressed  by  the  equa- 
tions. Unless  expressed  in  words,  the  ideas  are  liable  not  to 
be  expressed  at  all. 

The  equation  of  a  curve  is  an  equation  satisfied  by  the  co- 
ordinates of  every  point  of  the  curve  and  by  the  coordinates  of  no 
other  point. 

The  graph  of  an  equation  is  the  locus  of  a  point  whose  coordi- 
nates satisfy  the  equation. 

Exercises 

1.  Draw  and  discuss  the  following  loci: 

The  ordinate  of  any  point  of  a  certain  locus  is  twice  its  ab- 
scissa; the  X  of  every  point  of  a  certain  locus  is  half  its  y,  the  y  of 
a  point  is  1/3  of  its  x)  a  point  moves  in  such  a  way  that  its  lati- 
tude is  always  treble  its  longitude;  the  sum  of  the  latitude  and 
longitude  of  a  point  is  zero;  a  point  moves  so  that  the  difference 
in  its  latitude  and  longitude  is  always  zero. 

2.  Draw  this  locus:  Beginning  at  the  point  (1,  2),  a  point  moves 
so  that  its  gain  in  latitude  is  always  twice  as  great  as  its  gain  in 
longitude. 

3.  A  point  moves  so  that  its  latitude  is  always  greater  by  2  units 
than  three  times  its  longitude.  Write  the  equation  of  the  locus 
and  construct. 

4.  A  head  of  100  feet  of  water  causes  a  pressure  at  the  bottom  of 
43.43  pounds  per  square  inch.  Draw  a  locus  showing  the  relation 
between  head  and  pressure,  for  all  heads  of  water  from  0  to  200  feet. 

Suggestion:  There  are  several  ways  of  proceeding.  Let  pounds 
'per  square  inch  be  represented  by  abscissas  or  Xj  and  feet  of  water  be 
represented  by  ordinates  or  y.  Then  we  take  the  point  x  =  43.43, 
y  =  100  and  other  points,  as  a;  =  86.86,  y  =  200,  etc.,  and  draw  the 
line.     Otherwise  produce   the  equation  first  from   the  proportion 

100 
x:y:: 43.43  :  100,  or,  43.43?/  =  100 xory  =  v^^^  and  then  draw  the 

100 
graph  from  the  fact  that  the  latitude  is  always  ■  „--«  ^^  *^®  longitude. 

43.43 


38  ELEMENTARY  MATHEMATICAL  ANALYSIS         [§13 

Be  sure  that  the  scales  are  numbered  and  labeled  in  accordance  with 
suggestions  (4),  (5)  and  (6)  of  §12. 

5.  A  pressure  of  1  pound  per  square  inch  is  equivalent  to  a  column 
of  2.042  inches  of  mercury,  or  to  one  of  2.309  feet  of  water.  Draw  a 
locus  showing  the  relation  between  pressure  expressed  in  feet  of  water 
and  pressure  expressed  in  inches  of  mercury. 

Suggestion:  Let  x  =  inches  of  mercury  and  y  =  feet  of  water. 
First  properly  number  and  label  the  X-axis  to  express  inches  of  mer- 
cury and  number  and  label  the  F-axis  to  express  feet  ot  water.  Since 
negative  numbers  are  not  involved  in  this  exercise,  the  origin  may  be 
taken  at  the  lower  left-hand  corner  of  the  squared  paper.  First  locate 
the  point  x  =  2.042,  y  =  2.309  (which  are  the  corresponding  values 
given  by  the  problem)  and  draw  a  line  through  it  and  the  origin.  This 
is  the  required  locus  since  at  all  points  we  must  have  the  proportion 
X  :y  ::  2.042  : 2.309,  which  says  that  the  ordinate  of  every  point  ot  the 
locus  is  2309/2042  times  the  abscissa  of  that  point. 

6.  A  certain  mixture  of  concrete  (in  fact,  the  mixture  1  : 2  : 5)  con- 
tains 1.4  barrels  oi  cement  in  a  cubic  yard  of  concrete.  Draw  a  locus 
showing  the  cost  oi  cement  per  cubic  yard  of  concrete  for  a  range  ot 
prices  of  cement  from  $0.80  to  $2.00  per  barrel. 

Suggestion:  Let  x  be  the  price  per  barrel  of  cement  and  y  be  the 
cost  of  the  cement  in  1  cubic  yard  of  concrete.  Number  and  label 
the  two  scales  beginning  at  the  lower  left-hand  corner  as  origin.  Since 
prices  between  $0.80  and  $2.00  only  need  be  considered,  the  first 
division  on  the  X-axis  may  be  marked  $0.80  instead  of  0.  Each 
centimeter  may  represent  $0.10  on  each  scale.  The  cost  of  cement  per 
cubic  yard  of  concrete  must,  by  the  condition  of  the  problem,  be  1.4 
times  the  price  per  barrel  of  cement.  Hence  the  first  point  located 
on  the  vertical  scale  must  correspond  to  1.4  X  $0.80,  or  to  $1.12  cost 
per  cubic  yard.  As  this  is  the  lowest  cost  to  be  entered,  it  is  desirable 
not  to  start  the  vertical  scale  at  $0.00,  but  at  $1.00.  Thus  the  lower 
left-hand  corner  of  the  coordinate  paper  may  be  taken  as  the  point 
(0.80,  1.00)  in  a  system  in  which  the  unit  of  measure  is  1  cm.  =  10 
cents. 

7.  Draw  a  locus  showing  the  cost  per  cubic  yard  of  concrete  for 
various  prices  of  cement,  provided  $2.10  per  yard  must  be  added  to  the 
results  of  example  6  to  cover  cost  of  sand  and  crushed  stone. 

8.  Cast  iron  pipe,  class  A  (for  heads  under  100  feet),  weighs,  per 
foot  of  length:  4-inch,  20.0  pounds;  6-inch,  30.8  pounds;  8-inch,  42.9 
pounds.     For  each  size   of  pipe   construct   upon  a  single  sheet  of 


§141 


RECTANGULAR  COORDINATES 


39 


squared  paper  a  locus  showing  the  cost  per  foot  for  all  variations 
in  market  price  between  $20.00  and  $40.00  per  ton. 

Suggestion:  If  the  horizontal  scale  be  selected  to  represent 
price  per  ton,  the  scale  may  begin  at  20  and  end  at  40,  as  this  covers 
the  range  required  by  the  problem.  Therefore  let  1  cm.  represent 
$1.00.  Since  the  range  of  prices  is  from  1  cent  to  2  cents  per  pound, 
the  cost  per  foot  will  range  from  20  cents  to  40  cents  for  4-inch 
pipe  and  from  42.9  cents  to  85.8  cents  for  8-inch  pipe.  Hence 
for  the  vertical  scale  10  cents  may  be  represented  by  2  cm.  In  this 
case  the  vertical  scale  may  quite  as  well  begin  at  0  cents  instead  of 
at  20  cents,  as  there  is  plenty  of  room  on  the  paper. 


Y 

^J      <^       1             .5                     ^4 

irv  5          117    7 

\   1    \m=-2      .             /n»1.5^ 

r   \;    "         t    T 

A"-  2\             o       /      1   / 

^    ^      "^^           7 

\           \           o/              ym-H^l.5 

s  ^l    t^ 

-5X7 

\Z±Jn^ 

X           \  \r               X 

-5  -4       -3       -2    /-1\    /\     1         2         3        4         5 

/    y.Am-h 

J           /\      \ 

^71^ 

/           /-       -2.         ^ 

7           y           -Z^      ^ 

-/    7                \    \ 

L    J         Z^      \    \ 

7      t    Al        •    S^    \ 

r  J.           -^^^  W 

Fig.  27.— Lines  of  Slope  (1.5)  and  of  Slope  (-2). 


14.  Slope.  The  slope  of  a  straight  line  is  defined  to  be  the 
change  in  \j  for  an  increase  in  x  equal  to  1.  It  will  be  represented 
in  this  book  by  the  letter  m.  Thus  in  Fig.  27  the  line  A  has  the 
slope  m  =  1.5,  for  it  is  seen  that  at  any  point  of  the  line  the 
ordinate  y  gains  1.5  units  for  an  increase  of  1  in  x.  The  line  B, 
parallel  to  the  line  A,  is  also  seen  to  have  the  slope  equal  to  1.5. 
The  equation  of  the  line  A  is  obviously  y  =  1.5a:.  In  the  same 
figure  the  slope  of  the  line  C  is  —  2,  for  at  any  point  of  this  line 


40  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§15 

the  ordinate  y  loses  2  units  for  an  increase  in  x  equal  to  1.  The 
equation  of  the  Hne  C  is  obviously  y  =  —2x.  Line  D,  parallel  to 
line  C,  also  has  slope  ( —  2) 

If  h  be  the  change  in  y  for  an  increase  of  x  equal  to  k,  then  the 
slope  m  is  the  ratio  h  fk. 

The  technical  word  slope  differs  from  the  word  slope  or  slant  in 
common  language  only  in  the  fact  that  slope,  in  its  technical  use, 
is  always  expressed  as  a  ratio.  In  common  language  we  speak  of  a 
"slope  of  1  in  10,"  or  a  "grade  of  50  feet  per  mile,"  etc.  In  mathe- 
matics the  equivalents  are  "slope  =  1/10,"  "slope  =  50/5280," 
etc. 

As  already  indicated,  the  definition  of  slope  requires  us  to  speak 
in  mathematics  of  positive  slope  and  negative  slope.  A  line  of  pos- 
itive slope  extends  upward  with  respect  to  the  standard  direction 
OX  and  a  line  of  negative  slope  extends  downward  with  reference 
to  OX. 

In  a  similar  way  we  may  speak  of  the  slope  of  any  curve  at  a 
given  point  on  the  curve,  meaning  thereby  the  slope  of  the  tangent 
line  drawn  to  the  curve  at  that  point. 


Exercises 

1.  Give  the  slopes  of  the  lines  in  exercises  1  to  8  of  the  preceding 
set  of  exercises. 

^_  ^  ^  2x  X  X 

2.  Draw  y  =  x;  y  =  2x;  y  =  Sx;  y  =   3  5  2/  =  2'  ^  ""  4'  ^  ""  ~      ' 

y  =  —  Sx;  y  =  Ox. 

3.  Prove  that  y  =  mx  always  represents  a  straight  line,  no  matter 
what  value  m  may  have. 

15.  Equation  of  Any  Line.  Intercepts. — In  Fig.  28,  the  line 
MN  expresses  that  the  ordinate  y  is,  for  all  points  on  the  line,  always 
3  times  the  abscissa  x,  or  it  says  that  y  =  Sx.  The  line  HK  states 
that  "y  is  2  more  than  3x.''  Thus  the  hne  HK  has  the  equation 
y  =  Sx-i-2. 

In  general,  since  y  =  mx  is  always  a  straight  Une,^  then  y  = 
mx  -\-  b  is  a,  straight  Hne,  for  the  y  of  this  locus  is  merely,  in  each 
case,  the  y  of  the  former  increased  by  the  constant  amount  b  (which 

iSee  exercise  3,  §14,  above. 


§15] 


RECTANGULAR  COORDINATES 


41 


may,  of  course,  be  positive  or  negative).  Therefore,  y  =  mz  -\-  b 
is  a  line  parallel  to  y  =  mx.  The  distance  OB  (Fig,  28)  is  equal 
to  b.  The  distance  is  called  the  y-intercept  of  the  locus.  The 
distance  OA  is  equal  to  —  6  /m,  for  it  is  the  value  of  x  when  y  is 
zero.     It  is  called  the  x-intercept  of  the  locus. 


4        / 

K 

2 

3  / 

/' 

B 

1 

1    / 

/ 

-2 

A  / 

O 
-1 

. 

I             3 

/ 

2    / 

-2 

4 

/ 

-S 

J 

-4 

Fig.  28. — Intercepts. 


Exercises 

1.  Sketch,  from  inspection  of  the  equations,  the  lines  given  by : 


(a)  y  =  X. 

{h)  y  =  x  +  \. 

(c)   y  =x+2. 


(d)  2/  =  a;  +  3. 

(e)  y  =  X  -  \. 
if)   y  =x-2. 


42  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§16 

2.  Sketch,  from  inspection  of  the  equations,  the  lines  given  by : 

(a)  y  =  \x.  (J)  y  =  -  ix. 

(6)  y  =  ix.  (g)  y  =   -  X. 

(c)  y  =  X.  (h)  y  =  —  2x. 

(d)  y  =  2x.  (i)  y  =  -  3x. 

(e)  y  =  3x.  (j)  y  =  \/'2.r. 

3.  Sketch  the  lines  given  by : 

(a)  x  =  3.  (d)  y  =  1.  (g)  y    =  0. 

(6)  X  =  5.  (e)   y  =  5.  (h)  x    ^  0. 

(c)    X  =  -2.  (/)    y  =   -3.  (i)    X'  =  4. 

4.  Sketch  from  inspection  of  the  equations,  the  following : 

(a)  y  =  X  -{-  1. 

(6)  y  =  hx  -\-l. 

(c)  y  =  -2x  +  4. 

id)  y  =  5x-\-S. 

(e)  y  =  -5x  -2. 

5.  Sketch,  from  inspection  of  the  equations: 


(a) 

y  =x+4:. 

ib) 

y  -  2x  -  S  =  0. 

(c) 

y  +  lx  +  l/S  = 

0, 

(d) 

ax  -\-  by  =  c. 

(e) 

x/a  +  y/b  =  1. 

6.  The  shortest  distance  between  y  =  mx  and  y  =  mx  +  6  is  not  b. 
Show  that  it  equals  b/\/l  -\-  m*. 

16.  Additive  Properties.  Sometimes  a  useful  result  is  obtained 
by  adding  (or  subtracting)  the  corresponding  ordinates  of  two 
graphs.  Thus  in  Fig.  26,  operating  expenses  of  a  power  plant 
may  be  added  to  ordinates  representing  various  rates  of  divi- 
dends, and  compared  (by  subtraction)  with  monthly  revenue. 
Sometimes,  however,  it  becomes  necessary  to  determine  a  result 
by  adding  two  functions  corresponding  to  different  values  of  the 
variable  or  argument.  Fig.  29  is  an  excellent  illustration  of  this. 
This  diagram  enables  one  to  find  the  cost  of  a  cubic  j^ard  of 
"1:2:4"  concrete  (except  cost  of  mixing)  by  knowing  the  prices 
of  the  constituent  materials.     The  information  necessary  to  con- 


il6] 


RECTANGULAR  COORDINATES 


43 


struct  the  loci  is  given  in  the  first  line  of  Table  I,  p.  44.     The 

amount  of  cement  in  1  cubic  yard  of  1:2:4  concrete  is  seen  to  be 

1.58  barrels.     The  pricf  per  barrel  of  cement  may  be  considered 

a  variable  changing  with  the  condition  of  the  market  and  with 

the    locality    where    sold. 

Calling    Xi    the    price    of 

cement,  the  cost  yi  of  the 

cement  in   1  cubic  yard  of 

1:2:4  concrete  is  then,  for 

all    market    prices  of 

cement,  expressed  by  the 

equation: 


1   ^ 

Cost  of  each  Ingredient  i 

a  One 

J 

~~ 

"~ 

Cubic  Yard  of  1:2:4  Concrete 
for  Various  Prices  of  the 

Constituents               / 

y 

g 

f 

o 

o 
-g    4 

' 

/ 

>J 

/ 

o 

. 

s 

/ 

/ 

O 

S    3 

J 

/ 

f 

/ 

.f 

J 

/ 

.s 

^J 

€ 

y 

o' 

y 

^ 

1    2 

/ 

/ 

/' 

y 

/ 

/ 

■;f 

^ 

^ 

s 

/ 

•h 

/ 

ft»V 

2 

«? 

/ 

,* 

y 

^ 

i^ 

— 
_ 

-^ 

i 

t 

-<b 

0^ 

y 

'' 

y 

\ 

^ 

' 

s 

J 

4 

/ 

?,N^ 

> 

Q 

/ 

^ 

-" 

a 

/ 

/ 

^ 

n 

L 

^' 

_ 

J 

- 

1 

12  3  4 

Price  of  Stone  and  Sand  per  Cubic  Yard  and 
of  Cement  per  Barrel  in  Dollars 

Fia.  29. 


2/1  =  1.58a:i 

This  is  graphically  repre- 
sented in  Fig.  29  by  the 
line  of  slope  1.58.  Note  in 
this  case  that  the  slope 
of  the  line  has  a  "physi- 
cal" meaning,  namely  it  is 
the  co^i  of  the  cement  in  1 
cubic  yard  when  the  yrice 
is  $1.00  a  barrel.     In  the 

same  way  the  cost  of  the  sand  and  of  the  crushed  stone  in  1 
cubic  yard  of  concrete  for  various  market  prices  of  these  com- 
modities is  expressed  by  the  lines  of  Fig.  29  of  slopes  0.44  and 
0.88  respectively. 

Example:  Let  the  price  x\  of  cement  be  $1.20  per  barrel;  let 
the  price  a:2of  stone  be  $1.75  per  cubic  yard,  and  the  price  Xa  of  sand 
be  $1.10  per  cubic  yard.  Find  the  cost  of  the  materials  necessary 
to  make  1  cubic  yard  of  1:2:4  concrete.     Then,  from  Fig.  29 : 

xx  =  $1:20   then   ?/i  =  $1.90 
xi  =     1.75  2/2  =     1.54 

X3  =     1.10  2/3  =    0.48 

Total,  or  cost  of  material  for  1 

cubic  yard  of  concrete  =  $3.92 

The  cost  of  concrete,  y,  is  a  function  of  three  variables,  Xi,  X2, 


44 


ELEMENTARY  MATHEMATICAL  ANALYSIS 


§16 


xs,  all  of  which,  for  convenience'  sake,  have  been  measured  on  the 
same  scale  or  axis  OX.  The  representation  of  several  variables 
on  the  same  scale  need  not  cause  any  confusion. 

Since  in  this  case  the  prices  of  the  constituents  of  the  concrete 
are  not  the  same,  the  total  cost  of  1  cubic  yard  of  concrete  cannot 
be  found  by  adding  the  ordinates  at  the  same  abscissa  of  the 
three  graphs,  because  the  abscissas  or  the  various  market  prices 
of  the  ingredients  are  not  the  same. 

The  second  line  of  the  table  may  be  used  by  the  student  as  the 
basis  of  construction  of  another  diagram  similar  to  that  of  Fig.  29. 


TABLE  I 


The  quantities  of  material  required  to  make  1  cubic  yard  of  concrete 
(based  on  33 1  percent  voids  in  the  sand  and  45  percent  voids  in 
the  broken  stone). 


Mixture 

Quantities  of  materials  in  1  cubic  yard,  of  concrete 

Cement, 
barrels* 

Sand, 
cubic  yards 

Stone, 
cubic  yards 

1:2:4  concrete     

1.58 
1.33 

0.44 

0.46 

0.88 
0.92 

1:2^:5  concrete 

1:3:6  concrete 

*  A  barrel  (4  bags)  of  cement  weighs  380  pounds  and  contains  3|  cubic  feet  of 
cement. 

Note  :  The  student  may  be  interested  to  know  how  the  figures  in 
the  first  line  of  the  table  are  obtained.  The  explanation  will  best  be 
understood  if  the  figures  as  given  are  first  verified.  First  the  1.58 
barrels  of  cement  should  be  reduced  to  cubic  yards.  It  gives  0.22 
cubic  yard.  A  part  of  this  must  be  used  to  fill  the  33 1  percent  of 
voids  in  the  0.44  cubic  yard  of  sand.  The  cement  required  for  this  is 
0.146  cubic  yard.  Thus  the  sand  and  cement  combine  to  make 
0.44  +  0.22  -  0.146  or  0.514  of  mixed  material.  A  part  of  this  mix- 
ture is  used  to  fill  the  45  percent  voids  in  the  0.88  cubic  yard  of  stone, 
which  equals  0.396  cubic  yard.  Hence  the  total  volume  of  stone, 
sand  and  cement  is  0.88  +  0.514  —  0.396,  which  equals  0.998,  or  the 
cubic  yard  required. 

To  find  the  numbers  in  the  table,  the  above  process  needs  to  be 
reversed  and  stated  algebraically.  Thus,  to  make  a  cubic  yard  of 
1:2:4  concrete  let 


il6] 


RECTANGULAR  COORDINATES 


45 


X  =  cubic  yards  cement  required. 
y  =  cubic  yards  sand  required. 
z  =  cubic  yards  stone  required. 

Then,  from  the  given  porosities,  or  percent  of  voids, 

X   —  ^y  =  surplus  of  cement  after  filling  voids  in  sand. 
{x   —  ly)  -\-  y  =  volume  of  mixed  sand  and  cement. 
[(^  —  iy)  +  y]  —  0.452  =   surplus  of    mixed   sand    and    ce- 
ment after  filling  voids  in  stone . 
z  +  [{x  —  iy)  +  y]  —  0.452;  =  1,  the  total  volume, 
or, 

0.552  +  ly  +x  =^1 


M  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 

Cost  of  each  Ingredient  in  One  Cubic  Yard 
of  1:3:6  Concrete  for  Various  Prices 

of  the  Constituents                  > 

7 

I 

/ 

' 

^ 

/ 

/ 

y 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

y 

/ 

'o>^ 

^ 

0 

^\ 

/ 

!" 

^ 

y 

^/ 

/ 

y 

y 

/ 

/ 

i^ 

X 

** 

/ 

/ 

^* 

p 

/ 

X 

' 

/ 

X 

V 

' 

^/ 

^ 

L^ 

' 

^ 

\^ 

/ 

_J 

12  3  4 

Price  of  Stone  and  Sand  per  Cubic  Yard  and 
of  Cement  per  Basrel  iu  Dollars 

Fig.  30. 


Also,  because  the  mixture  is  1 : 2 : 4 : 

2x  =  y 
4x  =  z 
These  give: 

4.53a;  =  1 
X  =  0.22  cubic  yard 

Shorter  reasoning  is  as  follows:  As  the  voids  in  the  crushed  stone 
are  to  be  completely  filled  in  the  finished  concrete,  the  z  cubic  yards 


46  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§17 

of  stone  counts  as  only  0.55-2  cubic  yards  in  the  final  product.  As  the 
voids  in  the  sand  are  to  be  completely  filled  in  the  final  mixture,  the 
y  cubic  yards  of  sand  counts  as  only  ^-y  cubic  yards  in  the  final 
product.  As  there  are  no  voids  to  be  filled  in  the  cement,  it  counts 
as  X  cubic  yards  in  the  final  result.     Hence  the  equation 

^  +  ly  -{•  0.552  =  1,  etc. 


Exercise 

From  the  diagram.  Fig.  30,  determine  and  insert  in  a  table  like 
Table  I,  the  quantity  of  each  sort  of  material  in  1  cubic  yard  of 
1:3:6  concrete. 

THE  POWER  FUNCTION 

17.  Definition  of  the  Power  Function.  The  algebraic  function 
consisting  of  a  single  power  of  the  variable,  such  for  example  as  the 
functions  x^,  x^,  1  Ix,  l/x^,  x^^,  etc.,  stand  next  to  the  linear 
function  of  a  single  variable,  mx  -\-  h,  in  fundamental  impor- 
tance.    The  function  x"  is  known  as  the  power  function  of  x. 

18.  The  Graph  of  x^.  The  variable  part  of  many  functions  of 
practical  importance  is  the  square  of  a  given  variable.  Thus  the 
area  of  a  circle  depends  upon  the  square  of  the  radius;  the  distance 
traversed  by  a  faUing  body  depends  upon  the  square  of  the  elapsed 
time;  the  pressure  upon  a  flat  surface  exposed  directly  to  the  wind 
depends  upon  the  square  of  the  velocity  of  the  wind;  the  heat 
generated  in  an  electric  current  in  a  given  time  depends  upon  the 
square  of  the  number  of  amperes  of  current,  etc.,  etc.  Each  of 
these  relations  is  expressed  by  an  equation  of  the  form  y  =  ax^,  in 
which  X  stands  for  the  number  of  units  in  one  of  the  variable  quan- 
tities (radius  of  the  circle,  time  of  fall,  velocity  of  the  wind,  amperes 
of  current,  respectively,  in  the  above  named  cases)  and  in  which 
y  stands  for  the  other  variable  dependent  upon  these.  The  num- 
ber a  is  a  constant  which  has  a  value  suitable  to  each  particular 
problem,  but  in  general  is  not  the  same  constant  in  different  prob- 
lems. Thus,  if  y  be  taken  as  the  area  of  a  circle,  y  =  irx"^,  in  which 
X  is  the  radius  measured  in  feet  or  inches,  etc.,  and  y  is  measured  in 
square  feet  or  square  inches,  etc.;  or  if  s  is  the  distance  in  feet 
traversed  by  a  falling  body,  then  s  =  16.  U^,  where  t  stands  for  the 


§19]  RECTANGULAR  COORDINATES  47 

elapsed  time  in  seconds.  In  one  case  the  value  of  the  constant  a 
is  3.1416  and  in  the  other  its  value  is  16.1. 

Let  us  first  graph  the  abstract  law  or  equation  y  =  x"^,  in  which 
a  concrete  meaning  is  not  assumed  for  the  variables  x  and  y  but 
in  which  both  are  thought  of  as  abstract  variables.  First  form  a 
suitable  table  of  values  for  x  and  x^  as  follows: 

X  [-3  -2  -1  0  0.2  0.4  0.6  0.8  1.0  1.2  1.4  1.6  1.8  2  3 
a;2  or  2/1      9        4       10    0.04   0.16^0.36   0.64     1.0     1.44      1.96    2.56    3.24     4     9 

Here  we  have  a  series  of  pairs  of  values  of  x  and  y  which  are  asso- 
ciated by  the  relation  y  =  x"^.  Using  the  x  of  each  pair  of  values 
as  abscissa  with  its  corresponding  y  there  can  be  located  as  many 
points  as  there  are  pairs  of  values  in  the  table,  and  the  array  of 
points  thus  marked  may  be  connected  by  a  freely  drawn  curve. 
To  draw  the  curve  upon  coordinate  paper,  form  Ml,  the  origin 
may  be  taken  at  the  mid-point  of  the  sheet,  and  2  cm.  used  as  the 
unit  of  measure  for  x  and  y.  If  the  points  given  by  the  pairs  of 
values  are  not  located  fairly  close  together,  it  is  obvious  that  a 
smooth  curve  cannot  be  satisfactorily  sketched  between  the  points 
until  intermediate  points  are  located  by  using  intermediate  values 
of  X  in  forming  the  table  of  values.  The  student  should  think  of 
the  curve  as  extending  indefinitely  beyond  the  limits  of  the  sheet  of 
paper  used;  the  entire  locus  consists  of  the  part  actually  drawn  and 
of  the  endless  portions  that  must  be  followed  in  imagination  beyond 
the  range  of  the  paper.  If  the  graph  of  y  =  x^  be  folded  about 
the  7-axis,  OY,  it  will  be  noted  at  once  that  the  left  and  right 
portions  of  the  curve  will  exactly  coincide.  The  student  will 
explain  the  reason  for  this  fact. 

19.  Parabolic  Curves.  The  equations  y  =  x,  y  =  x^,  y  =  x^^ 
y  =  x^  should  be  graphed  by  the  student  on  a  single  sheet  of  coor- 
dinate paper,  using  2  cm.  as  the  unit  of  measure  in  each  case. 
Table  II  may  be  used  to  save  numerical  computation  in  the  con- 
struction of  the  graphs  of  these  power  functions.  As  in  the  case 
of  y  =  x\  a  smooth  curve  should  be  sketched  free-hand  through 
the  points  located  by  means  of  the  table  of  values,  and  intermediate 
values  of  x  and  y  should  be  computed  when  doubt  exists  in  the  mind 
of  the  student  concerning  the  course  of  the  curve  between  any  two 
points. 


48 


ELEMENTARY  MATHEMATICAL  ANALYSIS        [§19 


\ 

4 

»3 

/ 

\ 

3 

It  /»/ 

/ 

\ 

2 

/ 

\ 

1 

31^ 

^^^ 

-3         -5 

\ 

\P 

^ 

V       -1 

v 

J 

J            4 

4 

-2 

^ 

\ 

-3 

i 

-d 

^ 

a 

Fig.  31. — Parabolic  Curves. 


Fig.  32. — Graph  of  the   Power  Function  for  n  >  0    (Parabolic    Curves) 
in  the  First  Quadrant. 


|19] 


RECTANGULAR  COORDINATES 


49 


The  graphs  of  the  above  power  functions  are  observed  to  be 
continuous  lines,  without  breaks  or  sudden  jumps.     A  formal  proof 


TABLE  II 


X     \     x^ 

1 

x3 

Vx 

Vx 

^H 

1/x 

l/rc2 

0.2 

0.04 

0.008 

0.447 

0.585 

0.089 

5.000 

25.000 

0.4 

0.16 

0.064 

0.632 

0.737 

0.252 

2.500 

6.250 

0.6 

0.36 

0.216 

0.775 

0.843 

0.465 

1.667 

2.778 

0.8 

0.64 

0.512 

0.894 

0.928 

0.715 

1.250 

1.563 

1.0      1.00 

1.000 

1.000 

1.000 

1.000 

1.000 

1.000 

1.2j     1.44 

1.728 

1.095 

1.063 

1.312 

0.8333 

0.6944 

1.4j     1.96 

2.744 

1.183 

1.119 

1.657 

0.7143 

0.5102 

1.6 

2.56 

4.096 

1.265 

1.170 

2.034 

0.6250 

0.3906 

1.8 

3.24 

5.832 

1.342 

1.216 

2.415 

0.5556 

0.3086 

2.0 

4.00 

8.000 

1.414 

1.260 

2.828 

0.5000 

0.2500 

2.2 

4.84 

10.65 

1.483 

1.301 

3.263 

0.4545 

0.2066 

2.4 

5.76 

13.82 

1.549 

1.339 

3.717 

0.4167 

0.1736 

2.6 

6.76 

17.58 

1.612 

1.375 

4.193 

0.3846 

0.1479 

2.8 

7.84 

21.95 

1.673 

1.409 

4.685 

0.3571 

0.1276 

3.0 

9.00 

27.00 

1.732 

1.442 

5.196 

0.3333 

0.1111 

3.2 

10.24 

32.77 

1.789 

1.474 

5.724 

0.3125 

0.0977 

3.4 

11.56 

39.30 

1.844 

1.504 

6.269 

0.2941 

0.0865 

3.6 

12.96 

46.66 

1.897 

1.533 

6.831 

0.2778 

0.0772 

3.8 

14.44 

54.87 

1.949 

1.560 

7.407 

0.2632 

0.0693 

4.0 

16.00 

64.00 

2.000 

1.587 

8.000 

0.2500 

0.0625 

4.2!  17.64 

74.09 

2.049 

1.613 

8.608 

0.2381 

0.0567 

4.4    19.36 

85.18 

2.098 

1.639 

9.229 

0.2273 

0.0517 

4.6!  21.16 

97.34 

2.145 

1.663 

9.866 

0.2174  i 

0.0473 

4.8    23.04 

1 

110.6 

2.191 

1.687 

10.42 

0.2083  i 

0.0434 

1 
5.0   25.00 

125.0 

2.236 

1.710 

11.18 

0.2000  i 

0.0400 

5.2    27.04 

140.6 

2.280 

1.732  i 

11.85 

0.1923  ' 

0.0370 

5.4i  29.16 

157.5 

2.324  1 

1.754  i 

12.66     i 

0.1852  ; 

0.0343 

5.6    31.36 

175.6 

2.366 

1.776 

13.25 

0.1786 

0.0319 

5.8    33.64 

195.1 

2.408 

1.797 

13.97 

0.1724 

0.0297 

50 


ELEMENTARY  MATHEMATICAL  ANALYSIS 


l§19 


TABLE 

II. — {Continued) 

X 

a;2 

rc3 

Vx 

Vx 

x'^^ 

1/x 

1/x^ 

6.0 

36.00 

216.0 

2.449      1.817 

14.70 

0.1667 

0.0278 

6.2 

38.44 

238.3 

2.490      1.837 

15.44 

0.1613 

0.0260 

6.4 

40.96 

262.1 

2.530 

1.857 

16.19 

0.1563 

0.0244 

6.6 

43.56 

287.5 

2.569 

1.876 

16.96 

0.1515 

0.0230 

6.8 

46.24 

314.4 

2.608 

1.895 

17.33 

0.1471 

0:0216 

7.0 

49.00 

343.0 

2.646 

1.913 

18.52 

0.1429 

0.0204 

7.2 

51.84 

373.2 

2.683 

1.931 

19.32 

0.1389 

0.0193 

7.4 

54.76 

405.2 

2.720 

1.949 

20.13 

0.1351 

0.0183 

7.6 

57.76 

439.0 

2.757 

1.966 

20.95 

0.1316 

0.0173 

7.8 

60.84 

474.6 

2.793 

1.983 

21.79 

0.1282 

0.0164 

8.0 

64.00 

512.0 

2.828 

2.000 

22.63 

0.1250 

0.0156 

8.2 

67.24 

551.4 

2.864 

2.017 

23.48 

0.1220 

0.0149 

8.4 

70.56 

592.7 

2.898 

2.033 

24.35 

0.1190 

0.0142 

8.6 

73.96 

636.1 

2.933 

2.049 

25.22 

0.1163 

0.0135 

8.8 

77.44 

681.5 

2.966 

2.065 

26.11 

0.1136 

0.0129 

9.0 

81.00 

729.0 

3.000 

2.080 

27.00 

0.1111 

0.0123 

9.2 

84.64 

778.7 

3.033 

2.095 

27.91 

0.1087 

0.0118 

9.4 

88.36 

830.6 

3.066 

2.110 

28.82 

0.1064 

0.0113 

9.6 

92.16 

884.7 

3.098 

2.125 

29.74 

0.1042 

0.0109 

9.8 

96.04 

941.2 

3.130 

2.140 

30.68 

0.1020 

0.0104 

10.0 

100.00 

1000.0 

3.162 

2.154 

31.62 

0.1000 

i 

0.0100 

that  X"  is  a  continuous  function  for  any  positive,  rational  value  of 
n  will  be  given  later. 

All  of  the  graphs  here  considered  have  one  impor  tant  prop- 
erty in  common,  namely,  they  all  pass  through  the  points  (0,  0) 
and  (1, 1).  It  is  obvious  that  this  property  may  be  affirmed  of  any 
curve  of  the  class  y  =  a:",  if  n  is  a  positive  number.  These  curves 
are  known  collectively  as  curves  of  the  parabolic  family,  or  simply 
parabolic  curves.  The  curve  y  =  x"^  is  called  the  parabola. 
y  =  x^  is  called  the  cubical  parabola,  y  =  x'^  is  called  the  semi- 
cubical  parabola,  etc.  Curves  for  negative  values  of  n  do  not  pass 
through  the  point  (0,  0)  and  are  otherwise  quite  distinct.     They 


§20]  RECTANGULAR  COORDINATES  51 

are  known  as  curves  of  the  hyperbolic  type,  and  will  be  discussed 
later. 

The  student  should  cut  patterns  of  the  parabola,  the  cubical 
parabola  and  the  semi-cubical  parabola  out  of  heavy  paper  for 
use  in  drawing  these  curves  when  required.  Each  pattern  should 
have  drawn  upon  it  either  the  x-  or  y-axis  and  one  of  the  unit  lines 
to  assist  in  properly  adjusting  the  pattern  upon  squared   paper. 

20.  Symmetry.  In  geometry  a  distinction  is  made  between  two 
kinds  of  symmetry  of  plane  figures — symmetry  with  respect  to  a 
line  and  symmetry  with  respect  to  a  point.  A  plane  figure  is 
symmetrical  with  respect  to  a  given  line  if  the  two  parts  of  the 
figure  exactly  coincide  when  folded  about  that  line.  Thus  the  let- 
ters M  and  W  are  each  symmetrical  with  respect  to  a  vertical  line 
drawn  through  the  vertex  of  the  middle  angles.  We  have  already 
noted  that  y  =  x^  is  symmetrical  with  respect  to  OY. 

A  plane  figure  is  symmetrical  with  respect  to  a  given  point  when 
the  figure  remains  unchanged  if  rotated  180°  in  its  own  plane  about 
an  axis  perpendicular  to  the  plane  at  the  given  point.  Thus  the 
letters  N  and  Z  are  each  symmetrical  with  respect  to  the  mid-point 
of  their  central  line.  The  letters  H  and  0  are  symmetrical  both 
with  respect  to  lines  and  with  respect  to  a  point.  Which  sort  of 
symmetry  is  possessed  by  the  curve  y  =  x^'i     Why? 

Another  definition  of  symmetry  with  respect  to  a  point  is  per- 
haps clearer  than  the  one  given  in  above  statement :  A  curve  is 
said  to  be  symmetrical  with  respect  to  a  given  point  0  when  all 
lines  drawn  through  the  given  point  and  terminated  by  the  curve 
are  bisected  at  the  point  0. 

What  kind  of  symmetry  with  respect  to  one  of  the  coordinate 
axes  or  to  the  origin  (as  the  case  may  be)  does  the  point  (2, 3)  bear 
to  the  point  (-2,  3)?  To  the  point  (-2,-3)?  To  the  point 
(2,  -3)? 

Note  that  symmetry  of  the  first  kind  means  that  a  plane  figure  is 
unchanged  when  turned  180°  about  a  certain  line  in  its  plane,  and 
that  symmetry  of  the  second  kind  means  that  a  figure  is  unchanged 
when  turned  180°  about  a  certain  line  perpendicular  to  its  plane. 

21.  The  curves  in  the  diagram,  Fig.  31,  are  sketched  from  a 
limited  number  of  points  only,  but  any  number  of  additional 
values  of  x  and  y  may  be  tabulated  and  the  accuracy,  as  well  as 


52  ELEMENTARY  MATHEMATICAL  ANALYSIS         [§22 

the  extent,  of  the  graph  be  made  as  great  as  desired.  A  num- 
ber of  graphs  of  power  functions  are  shown  as  they  appear  in  the 
first  quadrant  in  Figs.  32  and  35.  The  student  should  explain  how 
to  draw  the  portions  of  the  curves  lying  in  the  other  quadrants 
from  the  part  appearing  in  the  first  quadrant. 

In  the  exercises  in  this  book  to  ''draw  a  curve"  means  to  con- 
struct the  curve  as  accurately  as  possible  from  numerical  or  other 
data.  To  ''sketch  a  curve"  means  to  produce  an  approximate  or 
less  accurate  representation  of  the  curve,  including  therein  its 
characteristic  properties,  but  without  the  use  of  extended  numer- 
ical data. 

Exercises 

1.  On  coordinate  paper  draw  the  curves  y  =  x^,  y  =  x^,  y  =  x^^, 
y  =  x^,  using  4  cm.  as  the  unit  of  measure.  On  the  same  sheet 
draw  the  lines  x  =  ±  1,  y  =  ±  1,  y  =  ±  x. 

2.  On  coordinate  paper  sketch  the  curves  x  =  y"^,  x  =  y^,  x  =  y^^^ 
X  =  y^.     Compare  with  the  curves  of  exercise  1. 

3.  Sketch  and  discuss  the  curves  y  =  "Sx,  y  =  \/a;,  y  =  \/x. 
Can  any  of  these  curves  be  drawn  from  patterns  made  from  the 
curves  of  exercise  1?  "Why?  Explain  the  graphs  of  the  first  and 
last  if  the  double  sign  "  +  "  be  understood  before  the  radicals,  and 
compare  with  the  graphs  when  the  positive  sign  only  is  to  be  under- 
stood before  the  radicals. 

4.  Draw  the  curve  y^  =  a;*.     Compare  with  the  curve  y  =  x^. 

5.  Name  in  each  case  the  quadrants  of  the  curves  of  exercises 
1-4,  and  state  the  reasons  why  each  curve  exists  in  certain  quad- 
rants and  why  not  in  the  other  quadrants. 

22.  Discussion  of  the  ParaboUc  Cixrves.  Draw  the  straight 
lines  X  =  l,x  =  —l,y=l,y=  — 1  upon  the  same  sheet  upon 
which  a  number  of  parabolic  curves  have  been  drawn.  These 
lines  together  with  the  coordinate  axes  divide  the  plane  into  a 
number  of  rectangular  spaces.  In  Fig.  33  these  spaces  are  shown 
divided  into  two  sets,  those  represented  by  the  cross-hatching, 
and  those  shown  plain.  The  cross-hatched  rectangular  spaces 
contain  the  lines  y  —  x  and  y=  —x  and  also  all  curves  of  the  para- 
bolic type.  No  parabolic  curve  ever  enters  the  rectangular  strips 
shown  plain  in  Fig.  33. 


§22] 


RECTANGULAR  COORDINATES 


53 


The  line  y  =  x  divides  the  spaces  occupied  by  the  parabolic 
curves  into  equal  portions.  Why  does  the  curve  y  =  x^  (in  the 
first  quadrant)  lie  below  this  line  in  the  interval  a;  =  0  to  a;  =  1, 
but  above  it  in  the  interval  to  the  right  of  a;  =  1?  On  the  other 
hand,  why  does  the  curve  y  =  \/x,  or  y^  =  x  (in  the  first  quad- 
rant), lie  above  the  line  y  =  xm  the  interval  x=Otox  =  land 
below  y  =  x  in  the  interval  to  the  right  of  a;  =  1? 

One  part  of  the  parabolic  curve  2/  =  x"  always  lies  in  the  first 
quadrant.     If  n  be  an  even  number,  another  part  of  the  curve  lies 


Fig.  33. — The  Regions  of  the  Parabolic  and  the  Hyperbolic  Curves. 
All  parabolic  curves  lie  within  the  cross-hatched  region.  All  hyperbolic 
curves  lie  within  the  region  shown  plain. 


in  which  quadrant?  If  n  be  an  odd  number,  the  curve  lies  in  which 
quadrants? 

If  the  exponent  n  of  any  power  function  be  a  positive  fraction, 
may  mjr,  the  equation  of  the  curve  may  be  written  y  =  x"^.  If 
in  this  case  both  m  and  r  be  odd,  the  curve  lies  in  which  quadrants? 
If  m  be  even  and  r  be  odd,  the  curve  lies  in  which  quadrants?  If 
m  be  odd  and  r  be  even,  the  curve  lies  in  which  quadrants?  If 
both  m  and  r  be  even  the  curve  lies  in  which  quadrants? 

A  curve  which  is  symmetrical  to  another  curve  with  respect  to  a 
line  may  be  spoken  of  figuratively  as  the  reflection  or  image  of  the 
second  curve  in  a  mirror  represented  by  the  given  line. 


54  ELEMENTARY  MATHEMATICAL  ANALYSIS  [§23 

Exercises 

Exercises  1-5  refer  to  curves  in  the  first  quadrant  only. 

1.  The  expressions  x"^,  a;^^,  x^,  x^  are  numerically  less  than  x  for 
values  of  x  between  0  and  1.  How  is  this  fact  shown  in  the  diagram, 
Fig.  31? 

2.  The  expressions  x^,  rc^^,  x"^,  x^  are  numerically  greater  than  x  for 
all  values  of  x  numerically  greater  than  unity.  How  is  this  fact 
pictured  in  the  diagram,  Fig.  31? 

3.  For  values  of  x  between  0  and  1,  x^  <  x^  K  x"^  <  x^'^^  <x. 
For  values  x  >  1,  x^  >  x^  >  x^  >a;^^2  >  x.  Explain  how  each  of  these 
facts  is  expressed  by  the  curves  of  Fig.  32. 

4.  Show  that  the  graphs  y  =  x^^,  y  =  x^^^,  y  =  x^-'^,  y  =  x^''^  are  the 
reflections  oi  y  =  a:%,  y  =  x^,  y  =  x^,  y  =  x^,  in  the  mirror  y  =  x. 

6.  Sketch  without  tabulating  the  numerical  values,  the  following 
loci:  y  =  x^°,  y  =  x^i,  y  =  x^oo,  y  =  x^-^K 

The  following  are  to  be  discussed  for  all  quadrants. 

6.  Sketch,  without  tabulating  numerical  values,  the  following  loci 
y'i  =  X*,  y^  =  x^,  y^  =  x^,  y^  =  x^,  y^  =  x^. 

7.  Sketch  the  following:  y^^  =  x^<^\  y^'^'-  =  x^\  i/^ooo  =  x^^^K 

8.  Sketch  the  following:  y  =  —  x^,  y  =  —  x^,  y^  =  —  x^. 

23.  HyperboUc  Type.  Loci  of  equations  of  the  form  ?/x«  =  1, 
or  y  =  Ifx",  where  n  is  positive,  have  been  called  hyperbolic 
curves.  The  fundamental  curve  xy  =  1,  or  y  =  1  /x  is  called  the 
rectangular  hyperbola.  Its  graph  is  given  in  Figs.  34  and  35, 
but  the  curve  should  be  drawn  independently  by  the  student,  using 
2  cm.  as  the  unit  of  measure.  Its  relation  to  the  x-  and  y-axes  is 
most  characteristic.  For  very  small  positive  values  of  x,  the  value 
of  y  is  very  large,  and  as  x  approaches  0,  y  increases  indefinitely. 
But  the  function  is  not  defined  for  the  value  x  =  0,  for  the  prod- 
uct xy  cannot  equal  1  if  a;  be  zero.  For  numerically  small  but 
negative  values  of  x,  y  is  negative  and  numerically  very  large,  and 
becomes  numerically  larger  as  x  approaches  0.  The  locus  thus 
approaches  indefinitely  near  to  the  F-axis,  as  x  approaches  zero. 

Instead  of  saying  that  "y  increases  in  value  without  limit,"  it 
is  equally  common  to  say  "y  becomes  infinite;"  in  fact,  '' infinite" 
is  merely  the  Latin  equivalent  of  "no  limit."  It  is  often  written 
?/  =  oo .     This  is  a  mere  abbreviation  for  the  longer  expressions, 


523] 


RECTANGULAR  COORDINATES 


55 


^'y  becomes  infinite"  or  "y  increases  in  value  without  limit." 
The  student  must  be  cautioned  that  the  symbol  <»  does  not  stand 


for  a  number,  and  that  "y 


must  not  be  interpreted  in  the 


same  way  that  "y  =  5"  is  interpreted. 

As  X  increases  from  numerically  large  negative  values  to  0, 
y  continually  decreases  and  becomes  negatively  infinite  (abbre- 
viated ?/  =  —  °° ) .     As  X  decreases  from  numerically  large  positive 


a 

Y 

I^ 

o 

Ih  Ij 

9 

\ 

r 

\ 

x^^'^ 

-^ 

0 

d 

^^2^1 

i-_ 

-3      -5 

I 

-1 

A 

1 

2           [ 

x^%=\ 

\- 

V 

I 

y' 

Fig.  34. — Hyperbolic  Curves. 


values  to  0,  y  continually  increases  and  becomes  infinite.  Thus, 
in  the  neighborhood  of  x  =  0,  ?/  is  discontinuous,  and,  in  this  case, 
the  discontinuity  is  called  an  infinite  discontinuity. 

On  account  of  the  symmetry  vnxy  =  \/\i  we  look  upon  x  as  a 
function  of  y,  all  of  the  above  statements  may  be  repeated,  merely 
interchanging  x  and  y  wherever  they  occur.  Thus,  there  is  an 
infinite  discontinuity  in  x,  as  y  passes  through  the  value  0. 

The  lines  XX'  and  YY'  which  these  curves  approach  as  near  as 


50 


ELEMENTARY  MATHEMATICAL  ANALYSLS 


l§23 


we  please,  but  never  meet,  are  called  the  asymptotes  of  the 
hyperbola. 

All  other  curves  of  the  hyperbolic  family,  such  as  yx"^  =  1, 
xy"^  =  1,  yH"^  =  1,  y^x^  =  1  and  the  like,  approach  the  X-  and 
F-axes  as  asymptotes.  The  rates  at  which  they  approach  the 
axes  depends  upon  the  relative  magnitudes  of  the  exponents  of  the 
powers  of  x  and  y,  the  quadrants  in  which  the  branches  lie  depend 
upon  the  oddness  or  evenness  of  these  exponents. 


■  '  '  I  I  '.y 


,r^\A'\<^'\i\rv}\}\^'\^'\o 


^^^ 

i 

y////////A2c -< 

y//y////////////yA 

— ■ 

r 



Fig.  35. — Hyperbolic  Curves  in  the  Fig.  36. — A  Hyperbola 
First  Quadrant,  y  =  l/a;i-*o8  is  the  Formed  by  Capillary  Action  of 
Adiabatic  Curve  for  Air.  Two  Converging  Plane  Plates. 


Exercises 


1.  Draw  accurately  upon  squared  paper  the  loci,  xy  =  1,  xy^  =  1, 
x^y  =  1,  xy^  =  1. 

2.  Show  that  the  curves  of  the  hyperbolic  type  lie  in  the  rectangular 
regions  shown  plain,  or  not  cross-hatched,  in  Fig.  33. 

3.  In  what  quadrants  do  the  branches  of  x^^y''  =  1  lie? 

4.  How  does  the  locus  of  x^y"^  =  1  differ  from  that  oi  xy  =  1? 

5.  Sketch,  showing  the  essential  character  of  each  locus,  the  curves 
x2t/3  =  1,  x^^y  =  1,  a;iooO|/  =  1. 

6.  Show  that  xy  =  a  passes  through  the  point  (\/a,  \/a);  that 
xy  =  a^  passes  through   (a,  a)  and  can  be  made  from  xy  =  1  hy 


§24]  RECTANGULAR  COORDINATES  57 

"stretching"  (if  a  >  1)  both  abscissas  and  ordinates  oi  xy  =  1  in  the 
ratio  1  :a.^ 

24.  Curves  Symmetrical  to  Each  Other.  Some  of  the  facts  of 
symmetry  respecting  two  portions  of  the  same  parabola  or  hyper- 
bola may  be  readily  extended  by  the  student  to  other  curves. 
First  answer  the  following  questions: 

How  are  the  points  {a,  h)  and  ( — a,  6)  related  to  the  F-axis? 
How  are  the  points  (a,  b)  and  {a,  —b)  related  to  the  X-axis? 
How  are  the  points  {a,  b)  and  (6,  o)  related  to  the  line  y  =  x? 
Prove  the  result  by  plane  geometry. 

The  following  may  then  be  readily  proved  by  the  student: 

Theorems  on  Loci 

I.  //  X  be  replaced  by  {—  x)  in  any  equation  containing  x  and  y, 
the  new  grarph  is  the  reflection  of  the  former  in  the  axis  YY\ 

IT.  If  y  be  replaced  by  {—y)in  any  equation  containing  x  and  y, 
the  new  graph  is  the  reflection  of  the  former  in  the  axis  XX'. 

III.  //  X  and  y  be  interchanged  in  any  equation  containing  x 
and  y,  the  new  graph  is  the  reflection  of  the  former  one  in  the  line 
y  =  X. 

25.  The  Variation  of  the  Power  Fimction.  The  symmetry  of 
the  graphs  of  the  power  function  with  respect  to  certain  lines  and 
points,  while  of  interest  geometrically,  nevertheless  does  not  con- 
stitute the  most  important  fact  in  connection  with  these  functions. 
Of  more  importance  is  the  law  of  change  of  value  or  the  law  by  which 
the  function  varies.  Thus  returning  to  a  table  of  values  for  the 
power  function  x^  for  the  first  quadrant, 

a;|0      1/2      1      3/2      2        5/2      3 


x^\0      1/4      1      9/4      4      25/4      9 

we  note  that  as  x  changes  from  0  to  1/2  the  function  grows  by  the 
small  amount  1/4.  As  a;  changes  from  1  /2  by  another  increment  of 
1/2  to  the  value  1,  the  function  increases  by  3/4  to  the  value  1. 
As  X  grows  by  successive  steps  or  increments  of  1  /2  unit  each,  it 
is  seen  that  x^  grows  by  increasingly  greater  and  greater  steps, 
until  finally  the  change  in  x^  produced  by  a  small  change  in  x 

iTo  "elongate  "  or  "stretch  "  in  the  ratio  2:  3  means  to  change  the  length 
of  a  line  segment  so  that  (original  length):   (new  or  stretched  length)  =  2: 3. 


58  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§26 

becomes  very  large.  Thus  the  step  by  step  increase  in  the  function 
is  a  rapidly  augmenting  one.  Even  more  rapidly  does  the  func- 
tion x^  gain  in  value  as  x  grows  in  value.  On  the  contrary,  for  posi- 
tive values  of  x  the  power  functions  1  /x,  1  /x^,  1  jx^,  etc.,  decrease 
in  value  as  x  grows  in  value.  Referring  to  the  definition  of  the 
slope  of  a  curve  given  in  §14,  we  see  that  the  parabolic  curves 
have  a  positive  slope  in  the  first  quadrant,  while  the  hyperbolic 
curves  have  always  a  negative  slope  in  the  first  quadrant. 

The  law  of  the  power  function  is  stated  in  more  definite  terms 
in  §34.  That  section  may  be  read  at  once,  and  then  studied 
a  second  time  in  connection  with  the  practical  work  which 
precedes  it. 

26.  Increasing  and  Decreasing  Functions.  As  a  point  passes 
from  left  to  right  along  the  X-axis,  x  increases  algebraically. 
As  a  point  moves  up  on  the  F-axis,  y  increases  algebraically  and 
as  it  moves  down  on  the  F-axis,  y  decreases  algebraically.  An 
increasing  function  of  x  is  one  such  that  as  x  increases  algebraically, 
y,  or  the  function,  also  increases  algebraically.  By  a  decreasing 
function  of  x  is  meant  one  such  that  as  x  increases  algebraically, 
y  decreases  algebraically.  Graphically,  an  increasing  function  is 
indicated  by  a  rising  curve  as  a  point  moves  along  it  from  left  to 
right.  The  power  function  y  =  x^  {n  positive)  is  an  increasing 
function  of  x  in  the  first  quadrant.  The  power  function  y  = 
x~'^  (—  n  negative)  in  the  first  quadi^ant  is  a  decreasing  function 
of  x. 

The  power  function  y  =  x^  is  an  increasing  function  for  all  values 
of  X  while  y  =  X-  issL  decreasing  function  in  the  second  quadrant 
but  an  increasing  function  in  the  first  quadrant.  In  a  case  like 
y  =  ±  a:^^,  where  y  has  two  values  for  each  positive  value  of  x,  it 
is  seen  that  one  of  these  values  increases  with  x  while  the  other 
decreases  with  x. 

Exercises 

1.  Consider  the  function  y  =  -{-  rc''^  and  construct  its  locus.  As  x 
grows  by  successive  steps  of  one  unit  each,  does  the  function  grow  by 
increasingly  greater  and  greater  steps  or  not?  Why?  Is  the  slope 
of  the  curve  an  increasing  or  a  decreasing  function  of  a;? 


§27J  RECTANGULAR  COORDINATES  59 

2.  Does  the  algebraic  value  of  the  slope  oi  xy  =  1  increase  with  x 
in  the  first  quadrant? 

3.  As  a:  changes  from  —  5  to  +5  does  the  slope  oi  y  =  x^  always 
increase  algebraically? 

4.  Express  in  the  language  of  mathematics  the  fact  that  the 
curves  y  =  a;",'  when  n  is  a  rational  number  greater  than  unity,  are 
concave  upward. 

Answer:  ''When  n  is  greater  than  unity,  the  slope  of  the  curve 
increases  as  x  increases." 

Express  in  a  similar  way  the  fact  that  the  curves  y  =  x^/"  are 
concave  downward. 

27.  The  Graph  of  the  Power  Function  when  x'*  has  a  Coeffi- 
cient.    If  numerical  tables  be  prepared  for  the  equations 

y    =  x^ 
and  y'  =  Sx^ 

then  for  like  values  of  x  each  ordinate,  y',  of  the  second  curve 
will  be  three  fold  the  corresponding  ordinate,  y,  of  the  first  curve. 
It  is  obvious  that  the  curve 

y'  =  ax''  (l) 

and  the  curve 

?/    =  a;"  (2) 

are  similarly  related;  the  ordinate  y'  of  any  point  of  the  first  locus 
can  be  made  from  the  corresponding  ordinate  y  {i.e.,  the  ordinate 
having  the  same  abscissa)  of  the  second  by  multiplying  the  latter 
by  a.  If  a  be  positive  and  greater  than  unity,  this  corresponds  to 
stretching  or  elongating  all  ordinates  of  (2)  in  the  ratio  1  :a;  if  a 
be  positive  and  less  than  unity,  it  corresponds  to  contracting  or 
shortening  all  ordinates  of  (2)  in  the  ratio  1 :  a. 

For  example,  the  graph  of  y'  =  ax"  can  be  made  from  the  graph 
of  y  =  x""  if  the  latter  be  first  drawn  upon  sheet  rubber,  and  if 
then  the  sheet  be  uniformly  stretched  in  the  y  direction  in  the  ratio 
I. a.  If  the  curve  be  drawn  upon  sheet  rubber  which  is  already 
under  tension  in  the  y  direction  and  if  the  rubber  be  allowed  to 
contract  in  the  y  direction,  the  resulting  curve  has  the  equation 
y  =  ax""  where  a  is  a  proper  fraction  or  a  positive  number  less  than 
unity. 

The  above  results  are  best  kept  in  mind  when  expressed  in  a 


60  ELEMENTARY  MATHEMATICAL  ANALYSIS         [§27 

slightly  different  from.  The  equation  y'  =  a-x"  can,  of  course,  be 
written  in  the  from  (2/Va)  =  a;«.  Comparing  this  with  the  equa- 
tion y  =  x"",  we  note  that  (y'/a)  =yory'  =  ay,  therefore  we  may 
conclude  generally  that  substituting  (y'/a)  for  y  in  the  equation  of 
any  curve  multiplies  all  of  the  ordinates  of  the  curve  by  a.  For 
example,  after  substituting  (y'/2)  for  y  in  any  equation,  the  new 
ordinate  y'  must  be  twice  as  large  as  the  old  ordinate  y,  in  order 
that  the  equation  remain  true  for  the  same  value  of  x. 

In  the  same  manner  changing  the  equation  y  =  x""  to  y  = 

—  j  ,  that  is,  substituting  {x'/a)  for  x  in  any  equation  multiplies 

all  of  the  abscissas  of  the  curve  by  a.  Multiplying  all  of  the  abscis- 
sas of  a  curve  by  a  elongates  or  stretches  all  of  the  abscissas  in 
the  ratio^  l:a  if  a  >  1,  but  contracts  or  shortens  all  of  the  abscis- 
sas if  a<l.  As  the  above  reasoning  is  true  for  the  equation  of 
any  locus,  we  may  state  the  results  more  generally  as  follows: 

Theorems  on  Loci 

IV.  Substituting  l-jfor  x  in  the  equation  of  any  locus  multiplies 
all  of  the  abscissas  of  the  curve  by  a. 

V.  Substituting  l  —  j  for  y  in  the  equation  of  any  locus  multiplies  all 

of  the  ordinates  of  the  curve  by  a. 

Note:  It  is  not  necessary  to  retain  the  symbols  x'  and  y'  to 
indicate  new  variables,  if  the  change  in  the  variable  be  otherwise 
understood. 

Exercises 

1.  Without  actual  construction,  compare  the  graphs  y  =  x^  and 

a;2  1  2 

y  =  5x2;    y  =  x^  and  y  =  ^;    y=  —  and  y  =    -;    y=x^  and  y  =  2x^; 
^  x  X 

y  =  x^^  and  y  =  g  • 

2.  Without  actual  construction,  compare  the  graphs  y  =  x^  and 
2/ =  U)   '   2/=^'and2  =  a;3;    y=x^a.ndy  =  U]   ;    y=x^and-^  =  x^. 

iSee  footnote,  p.  57. 


528] 


RECTANGULAR  COORDINATES 


61 


y^  =  a;' 


3.  Compare  y"^  =  x^  and  y^  =   (2)  ;  y^  =  x'and  Igj    = 

28.  Orthographic  Projection.  In  elementary  geometry  we 
learned  that  the  projection  of  a  given  point  P  upon  a  given  line  or 
plane  is  the  foot  of  the  perpendicular  dropped  from  the  given  point 
upon  the  given  line  or  plane.  Likewise  if  perpendiculars  be 
dropped  from  the  end  points  A  and  B  of  any  line  segment  AB  upon 
a  given  line  or  plane,  and  if  the  feet  of  these  perpendiculars  be 
called  P  and  Q,  respectively,  then  the  line  segment  PQ  is  called  the 
projection  of  the  line  AB.  Also,  if  perpendiculars  be  dropped 
from  all  points  of  a  given  curve  AB  upon  a  given  plane  MN,  the 


Fig.  37. — Orthographic  Projection  of  Line  Segments 


locus  of  the  feet  of  all  of  the  perpendiculars  so  drawn  is  called  the 
projection  of  the  given  curve  upon  the  plane  MN. 

To  emphasize  the  fact  that  the  projections  were  made  by  using 
perpendiculars  to  the  given  plane,  it  is  customary  to  speak  of  them 
as  orthogonal  or  orthographic  projections. 

The  shadow  of  a  hoop  upon  the  ground  is  not  the  orthographic 
projection  of  the  hoop  unless  the  rays  of  light  from  the  sun  strike 
perpendicular  to  the  ground.  This  would  only  happen  in  our  lat- 
itude upon  a  non-horizontal  surface. 

The  shortening  by  a  given  fractional  amount  of  all  of  a  set  of 


62  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§29 

parallel  line  segments  of  a  plane  may  be  brought  about  geometric- 
ally by  orthographic  projection  of  all  points  of  the  line  segments 
upon  a  second  plane.  For,  in  Fig.  37,  let  Ai^i,  A2B2,  A^iBs, 
etc.,  be  parallel  line  segments  lying  in  the  plane  MN.  Let  their 
projections  on  any  other  plane  be  A'lC'i,  A'^d,  A\C''6,  etc., 
respectively.  Draw  A2C2  parallel  to  A '2^2  and  Aid  parallel  to 
A\C'i,  etc.  Then  since  the  right  triangles  AiBid,  A2B2C2, 
AzBzCs,  etc.,  are  similar, 

A  1^1  ^  A2B2  ^  AsBj 

Aid      A2C2      ~A,Cz 

Call  this  r^io  a.  It  is  evident  that  a>l.  Substitute  the 
equals:  A'lC'i  =  AiCi,    A^C'i  =  A2C2,  etc.     Then: 

AxBi   _  A2B2  _  AS  3  _  _  a 

A\C\  ~  Z^C^  ~  Z^C^s  ~  '   '   '  ~  I 

The  numerators  are  the  original  line  segments;  the  denominators 
are  their  projections  on  the  plane  MO.  The  equality  of  these 
fractions  shows  that  the  parallel  lines  have  all  been  shortened  in 
the  ratio  a  :1. 

The  above  work  shows  that  to  produce  the  curve  y  =  (x/a)'', 
(a  <  1),  from  y  ==  x^hy  orthographic  projection  it  is  merely  neces- 
sary to  project  all  of  the  abscissas  of  y  =  x""  upon  a  plane  passing 
through  YOY'  making  an  angle  with  OX  such  that  unity  on  OX 
projects  into  a  length  a  on  the  projection  of  OX.  To  produce 
the  curve  y  =  ax""  (a  <1)  from  2/  =  x"  by  orthographic  projection 
it  is  merely  necessary  to  project  all  of  the  ordinates  of  ?/  =  x"  upon 
a  plane  passing  through  XOX'  making  an  angle  with  OY  such  that 
unity  on  OF  projects  into  the  length  a  on  the  projection  of  OY. 

To  lengthen  all  ordinates  of  a  given  curve  in  a  given  ratio, 
1  :  a,  the  process  must  be  reversed;  that  is,  erect  perpendiculars  to 
the  plane  of  the  given  curve  at  all  points  of  the  curve,  and  cut  them 
by  a  plane  passing  through  XOX'  making  an  angle  with  OY  such 
that  a  length  a  (a>l)  measured  on  the  new  7-axis  projects  into 
unity  on  OF  of  the  original  plane. 

29.  Change  of  Unit.  To  produce  the  graph  of  y  =  lOx^  from 
that  of  2/  =  x"^,  the  stretching  of  the  ordinates  in  the  ratio  1 :  10 
need  not  actually  be  performed.     If  the  unit  of  the  vertical  scale 


§30] 


RECTANGULAR  COORDINATES 


63 


1.0 


oi  y  =  x^  be  taken  1  /lO  of  that  of  the  horizontal  scale,  and  the 
proper  numerical  values  be  placed  upon  the  divisions  of  the 
scales,  then  obviously  the  graph  oi  y  =  x^  may  be  used  for  the 
graph  of  2/  =  lOx^.  Suitable  change  in  the  unit  of  measure  on  one 
or  both  of  the  scales  oi  y  =  x""  is  often  a  very  desirable  method  of 
representing  the  more  general  curve  y  =  ax"". 

An  interesting  example  is  given  in  Fig.  38.  The  period  of  vi- 
bration of  a  simple  pendulum  is  given  by  the  formula  T  =  Tr\/l/g. 
When  g  =  981  cm.  per 
second  per  second  (abbre- 
viated cm. /sec. 2)  this  gives 
T  =  0.1003Vr>  which  for 
many  purposes  is  suffici- 
ently accurate  when  writ- 
ten T  =  0.10\/1.  In  this 
equation  T  must  be  in  sec- 
onds and  I  in  centimeters. 
Thus  when  I  =  100  cm.,  T 
=  1  sec,  so  that  the  graph 
may  be  made  by  drawing 
the  parabola  y  =  \/x  from 
the  pattern   previously 

made  and  then  attaching  the  proper  numbers  to  the  scales,  as 
shown  in  Fig.  38. 

30.  Variation.  The  relation  between  y  and  x  expressed  by  the 
equation  y  =  ax",  where  n  is  any  positive  number,  is  often  expressed 
by  the  statement  "?/  varies  as  the  nth  power  of  x,"  or  by  the 
statement  "?/  ^s  proportional  to  x»."  Likewise,  the  relation 
y  =  a/x",  where  n  is  positive,  is  expressed  by  the  statement 
"?/  varies  inversely  as  the  nth  power  of  x."  The  statement  "the 
elongation  of  a  coil  spring  is  proportional  to  the  weight  of  the  sus- 
pended mass"  tells  us: 

y  =  mx  (1) 

where  y  is  the  elongation  (or  increase  in  length  from  the  natural 
or  unloaded  length)  of  the  spring,  and  x  is  the  weight  suspended  by 
the  spring,  but  it  does  not  give  us  the  value  of  m.  The  value  of  m 
may  readily  be  determined  if  the  elongation  corresponding  to  a 
given  weight  be  given.     Thus  if  a  weight  of  10  pounds  when  sus- 


10.8 

0.4 
0.2 


0  20  40  60  80  100  120  140  160 180  200 
Xength  in  Cm.. 

FiG.  38. — Relation  of  Length  of  a  Simple 
Pendulum  to  Period  of  Vibration. 


64  ELEMENTARY  MATHEMATICAL  ANALYSIS         [§30 

pended  from  the  spring  produces  an  elongation  of  2  inches  in  the 
length  of  the  coil,  then,  substituting  a:  =  10  and  y  =  2  in  (1), 

2  =  mlO 
and  hence  m  =  1/5 

If  this  spring  be  used  in  the  construction  of  a  spring  balance,  the 
length  of  a  division  of  the  uniform  scale  corresponding  to  1  pound 
will  be  1  /5  inch. 
A  special  symbol,   ex  ,  is  often  used  to  express  variation.     Thus 

y  ^  lid-' 

states  that  y  varies  inversely  as  d^.     it  is  equally  well  expressed  by : 

h 

where  ^  is  a  constant  called  the  proportionality  factor. 

The  statements  "?/  varies  jointly  as  u  and  v,"  and  ''y  varies 
directly  as  u  and  inversely  as  y,"  mean,  respectively: 

y  =  auv 

au 

y  =  — 

V 

Thus  the  area  of  a  rectangle  varies  jointly  as  its  length  and  breadth, 
or, 

A  =  kLB 

If  the  length  and  breadth  are  measured  in  feet  and  A  in  square  feet, 
k  is  unity.  But,  if  L  and  B  are  measured  in  feet  and  A  in  acres, 
then  k  =  1  /43560.  If  L  and  B  are  measured  in  rods  and  A  in 
acres,  then  A;  =  1  /160. 

From  Ohm's  law,  we  say  that  the  electric  current  in  a  circuit 
varies  directly  as  the  electromotive  force  and  inversely  as  the 
resistance,  or: 

C  ^  E/R  or  (7  =  kE/R 

The  constant  multiplier  is  unity  if  C  be  measured  in  amperes,  E 
in  volts,  and  R  in  ohms,  so  that  for  these  units 

C  =  E/R 


§31]  RECTANGULAR  COORDINATES  65 

31.  Illustrations  from  Science.  Some  of  the  most  important 
laws  of  natural  science  are  expressed  by  means  of  the  power  func- 
tion^ or  graphically  by  means  of  loci  of  the  parabolic  or  hyperbolic 
type. 

The  linear  equation  y  =  mx  is,  of  course,  the  simplest  case  of  the 
power  function  and  its  graph,  the  straight  line,  may  be  regarded  as 
the  simplest  of  the  curves  of  the  parabolic  type.  The  following 
illustrations  will  make  clear  the  importance  of  the  power  function 
in  expressing  numerous  laws  of  natural  phenomena.  Later  the 
student  will  learn  of  two  additional  types  of  fundamental  laws  of 
science  expressible  by  two  functions  entirely  different  from  the 
power  function  now  being  discussed. 

The  instructor  will  ask  oral  questions  concerning  each  of  the 
following  illustrations.  The  student  should  have  in  mind  the 
general  form  of  the  graph  in  each  case,  but  should  remember  that 
the  law  of  variation,  or  the  law  of  change  of  value  which  the  func- 
tional relation  expresses,  is  the  matter  of  fundamental  importance. 
The  graph  is  useful  primarily  because  it  aids  to  form  a  mental  pic- 
ture of  the  law  of  variation  of  the  function.  The  practical  graph- 
ing of  the  concrete  illustrations  given  below  will  not  be  done  at 
present,  but  will  be  taken  up  later  in  §33. 

(a)  The  pressure  of  a  fluid  in  a  vessel  may  be  expressed  in  either 
pounds  per  square  inch  or  iu  terms  of  the  height  of  a  column  of 
mercury  possessing  the  same  static  pressure.    Thus  we  may  write: 

p  =  0.492/1  (1) 

in  which  p  is  pressure  in  pounds  per  square  inch  and  h  is  the  height 
of  the  column  of  mercury  lq  inches.  The  graph  is  the  straight 
line  through  the  origin  of  slope  492  /lOOO.  The  constant  0.492  can 
be  computed  from  the  data  that  the  weight  of  mercury  is  13.6  times 
that  of  an  equal  volume  of  water  and  that  1  cubic  foot  of  water 
weighs  62.5  pounds. 

In  this  and  the  following  equations,  it  must  be  remembered 
that  each  letter  represents  a  riumber,  and  that  no  equation  can 
be  used  until  all  the  magnitudes  involved  are  expressed  in  terms 
of  the  particular  units  which  are  specified  in  connection  with 
that  equation. 

» For  brevity  ax"  as  well  as  x"  will  frequently  be  called  a  power  function  of  x. 


66  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§31 

(6)  The  velocity  of  a  falling  body  which  has  fallen  from  a  state 
of  rest  during  the  time  t,  is  given  by 

V  =  32.2^  (2) 

in  which  t  is  the  time  in  seconds  and  v  is  the  velocity  in  feet 
per  second.     If  t  is  measured  in  seconds  and  v  is  in  centimeters  per 
second,  the  equation   becomes^    v  =  981^.     In   either  case  the 
graph  is  a  straight  line,  but  the  lines  have  different  slopes, 
(c)  The  space  traversed  by  a  falling  body  is  given  by 

s  =  kt'  .  (3) 

or,  in  English  units  (s  in  feet  and  t  in  seconds) : 

s  =  16.1^2  (4) 

{d)  The  velocity  of  the  falling  body,  from  the  height  h  is: 

V  =  \/2gh  =  VoiAh  (5) 

The  resistance  of  the  air  is  not  taken  into  account  in  formulas 
(2)  to  (5). 
The  formula  equivalent  to  (5) : 

|wy2  =  mgh  (6) 

where  m  is  the  mass  of  the  body,  expresses  the  equivalence  of 
^mv^,  the  kinetic  energy  of  the  body,  and  mgh,  the  work  done 
by  the  force  of  gravity  mg,  working  through  the  distance  h. 

1  A  full  discussion  of  the  process  of  changing  formulas  like  the  ones  in  the  pres- 
ent section  into  a  new  set  of  units  should  be  sought  in  text-books  on  physics  and 
mechanics.  The  following  method  is  sufficient  for  elementary  purposes.  First, 
write  (for  the  present  example)  the  formula  v  =  32.2  t  where  v  is  in  ft./sec.  and 
t  is  in  seconds.  For  any  units  of  measure  that  may  be  used,  there  holds  a  general 
relation  v  =  ct,  where  c  is  a  constant.  To  determine  what  we  may  call  the 
dimensions  of  c,  substitute  for  all  letters  in  the  formula  the  names  of  the  units  in 
which  they  are  expressed,  treating  the  names  as  though  they  were  algebraic 
numbers.  From  v  =  ct  write,  ft./sec.  =  c  sec.  Hence  (solving  for  dimensions  of  c), 
c  has  dimensions  ft. /sec. 2  Thereiore  in  the  given  case,  we  know  c  =  32.2  ft./sec^. 
To  change  to  any  other  units  simply  substitute  equals  for  equals.  Thus  1  ft.  = 
30.5  cm.,  hence  c  =  32.2  X  30.5  cm./sec.2  =  981  cm./sec* 

To  change  velocity  from  mi./hr.  to  ft./sec.  in  formula  (19)  below,  we  have 
R  =  0.003  F2  where  R  is  in  Ib./sq.  ft.  and  V  is  in  mi./hr.  Write  the  general 
formula  R  =  cV^.  The  dimensions  of  c  are  (lb.  /  ft.2)  -=-  (mi.Vhr.2)  or  (lb.  /  ft. 2)  X 
(hr.Vnii.*).  In  the  given  case  we  have  the  value  of  c  =  0.003  (Ib./tt.^)  X 
(hr.Vmi^).  To  change  V  to  ft./sec,  substitute  equals  for  equals,  namely  1  hr.=» 
3600  sec,  1  mi.=  5280  ft.,  or  merely  (approximately)  mi./hr.  =  f  ft./sec. 


§31]  RECTANGULAR  COORDINATES  67 

(e)  The  intensity  of  the  attraction  exerted  on  a  unit  mass  by  the 
sun  or  by  any  planet  varies  inversely  as  the  square  of  the  distance 
from  the  center  of  mass  of  the  attracting  body.  If  r  stand  for 
that  distance  and  if  /  be  the  force  exerted  on  unit  mass  of  the 
attracted  body,  then 

■m 
^       r'  (7) 

The  constant  m  is  the  value  of  the  force  when  r  is  unity. 

(/)  The  forniula  for  the  horse  power  transmissible  by  cold-rolled 
shafting  is: 

where  E  is  the  horse  power  transmitted,  d  the  diameter  of  the 
shaft  in  inches,  and  N  the  number  of  revolutions  per  minute. 

The  rapid  variation  of  this  function  (as  the  cube  of  the  diameter) 
accounts  for  some  interesting  facts.  Thus  doubling  the  size  of  the 
shaft  operating  at  a  given  speed  increases  8-fold  the  amount  of 
power  that  can  be  transmitted,  while  the  weight  of  the  shaft  is 
increased  but  4-fold. 

If  H  be  constant,  N  varies  inversely  as  d^.  Thus  an  old-fash- 
ioned 50-h.p.  overshot  water-wheel  making  three  revolutions  per 
minute  requires  about  a  9-inch  shaft,  while  a  DeLaval  50-h.p. 
steam  turbine  making  16,000  revolutions  per  minute  requires  a 
turbine  shaft  but  little  over  1  /2  inch  in  diameter. 

{g)  The  period  of  the  simple  pendulum  is 

T  =  Wl]g  (9) 

where  T  is  the  time  of  one  swing  in  seconds,  I  the  length  of  the 
pendulum  in  feet  and  g  =  32.2  ft./sec.,^  approximately. 

{h)  The  centripetal  force  on  a  particle  of  weight  W  pounds, 
rotating  in  a  circle  of  radius  R  feet,  at  the  rate  of  N  revolutions 
per  second  is 

F^'^^WRNl  (10) 

or,  if  fir  =  32. 16  ft./sec.2, 

F  =  l221QWRm  (11) 


68  ELEMENTARY  MATHEMATICAL  ANALYSIS         [§31 

where  F  is  measured  in  pounds.  If  N  be  the  number  of  revolutions 
per  minute,  then 

^  SQOOg  ^^^^ 

=  0.000341  TF72iV2  (13) 

(i)  An  approximate  formula  for  the  indicated  horse  power 
required  for  a  steamboat  is: 

LH.P.=  ^^  (14) 

where  S  is  speed  in  knots,  D  is  displacement  in  tons,  and  C  is  a  con- 
stant appropriate  to  the  size  and  model  of  the  ship  to  which  it  is 
applied.  The  constant  ranges  in  value  from  about  240,  for  finely 
shaped  boats,  to  200,  for  fairly  shaped  boats. 

(j)  Boyle's  law  for  the  expansion  of  a  gas  maintained  at 
constant  temperature  is 

pv  =  C  (15) 

where  p  is  the  pressure  and  v  the  volume  of  the  gas,  and  C  is  a  con- 
stant. Since  the  density  of  a  gas  is  inversely  proportional  to  its 
volume,  the  above  equation  may  be  written  in  the  form 

p  =  cp  (16) 

in  which  p  is  the  density  of  the  gas. 

(k)  The  flow  of  water  over  a  trapezoidal  weir  is  given  by 

q  =  3.S7Lh"^  (17) 

where  q  is  the  quantity  in  cubic  feet  per  second,  L  is  the  length  of 
the  weir^  in  feet  and  h  is  the  head  of  water  on  the  weir,  in  feet. 

(I)  The  physical  law  holding  for  the  adiabatic  expansion  of 
air,  that  is,  the  law  of  expansion  holding  when  the  change  of 
volume  is  not  accompanied  by  a  gain  or  loss  of  heat,^  is 
expressed  by 

p  =  cp'-'^^  (18) 

1  The  instructor  is  expected  fully  to  explain  the  meaning  of  the  technical  terms 
here  used. 

2  Note  that  when  a  vessel  containing  a  gas  is  insulated  by  a  non-conductor  of 
heat,  so  that  no  heat  can  enter  or  escape  from  the  vessel,  that  the  temperature  of 
the  gas  will  rise  when  it  is  compressed,  or  fall  when  it  is  expanded.  Adiabatic  expan- 
sion may  be  thought  of,  therefore,  as  taking  place  in  an  insulated  vessel. 


§31]  RECTANGULAR  COORDINATES  69 

This  is  a  good  illustration  of  a  power  function  with  fractional  expo- 
nent. The  graph  is  not  greatly  different  from  the  semi-cubical 
parabola 

y  =  cx"^ 
(m)  The  pressure  or  resistance  of  the  air  upon  a  flat  surface  per- 
pendicular to  the  current  is  given  by  the  formula 

n  =  0.00372  (19) 

in  which  F  is  the  velocity  of  the  air  in  miles  per  hour  and  R  is  the 
resulting  pressure  upon  the  surface  in  pounds  per  square  foot. 
According  to  this  law,  a  20-mile  wind  would  cause  a  pressure  of 
about  1.2  pounds  per  square  foot  upon  the  flat  surface  of  a  building. 
One  foot  per  second  is  equivalent  to  about  2  /3  mile  per  hour,  so 
that  the  formula  when  the  velocity  is  given  in  feet  per  second 
becomes: 

R  =  0.001372  (20) 

(n)  The  power  used  to  drive  an  aeroplane  may  be  divided  into 
two  portions.  One  portion  is  utilized  in  overcoming  the  resistance 
of  the  air  to  the  onward  motion.  The  other  part  is  used  to  sustain 
the  aeroplane  against  the  force  of  gravity.  The  first  portion  does 
"useless"  work — work  that  should  be  made  as  small  as  possible  by 
the  shapes  and  sizes  of  the  various  parts  of  the  machine.  The 
second  part  of  the  power  is  used  to  form  continuously  anew  the 
wave  of  compressed  air  upon  which  the  aeroplane  rides.  Calling 
the  total  power ^  P,  the  power  required  to  overcome  the  resistance 
Pr,  and  that  used  to  sustain  the  aeroplane  P„  we  have 

P  =  Pr  +  P.  (21) 

We  learn  from  the  theory  of  the  aeroplane  that  Pr  varies  as  the 
cube  of  the  velocity,  while  P,  varies  inversely  as  7,  so  that 

Pr  =  cV^  (22) 

and 

P.  =    y  (23) 

Thus  at  high  velocity  less  and  less  power  is  required  to  sustain  the 
aeroplane  but  more  and  more  is  required  to  meet  the  friction  al 

1  Power  (  =  work  done  per  unit  time)  is  measured  by  the  unit  horse  power,  which  is 
550  foot-pounds  per  second. 


70 


ELEMENTARY  MATHEMATICAL  ANALYSIS 


[§31 


resistance  of  the  medium.  The  law  expressed  by  (23)  that  less 
and  less  power  is  required  to  sustain  the  aeroplane  as  the  speed  is 
increased  is  knOwn  as  Langley's  Law.  From  this  law  Langley  was 
convinced  that  artificial  flight  was  possible,  for  the  whole  matter 
seemed  to  depend  primarily  upon  getting  up  sufficient  speed.  It 
is  really  this  law  that  makes  the  aeroplane  possible.  An  analogous 
case  is  the  well-known  fact  that  the  faster  a  person  skates,  the 
thinner  the  ice  necessary  to  sustain  the  skater.     In  this  case 


S        M-    J    T    J-T  4    4    -f  J- 

S     jf  f  ^-i  /  t  tj^yz 

S     ^t  Tt  t^^4  J  T-J A 

oi     jt  -#  ul  ixr^^^^  I 

^J       ^ZlMlJ   Lj.    tj^YjlA 

?Q     -LI4  Li  l/  ///^// 

is     4  tt-Tti  /-/-yV%^ 

1?     7  ^rl4  ti.^niAQi./^ 

fi     Tli^ll-?-A7^  Z^Z/!^^^^^ 

f,      O^-v'-^^Z^Z^^i^^^^^ 

Z          ttttJj-4J~^Kt^^^ 

t          V^Trft^t/7Y/7y7yY. 

9    -iJ-J7ZZZfe?^^i22^^^ 

I    44fl^i55?^^^^?^^ 

in    \\^l  nr//^^'^^/A^-<' 

I  \\\N/fj//M'yyy^^ 

\   /#^^^pp^ 

/  If//////  /)yy/^^y^ 

li^m//m^^ 

\m//^w^ 

3    A       -^^^ 

I      >  ^                           1 

^\^^ 

S  8 


8  888888  8888  S§  8: 


Gals. for  One  Foot  Depth 


Fig.   30. — Capacity  of  Rectangular  and  Circular  Tanks  per  Foot  of  Depth. 


part  of  the  energy  of  the  skater  is  continually  forming  anew  on 
the  thin  ice  the  wave  of  depression  which  sustains  the  skater, 
while  the  other  part  overcomes  the  frictional  resistance  of  the 
skates  on  the  ice  and  the  resistance  of  the  air. 

(0)  The  capacity  of  cast-iron  pipe  to  transmit  water  is  often 
given  by  the  formula: 


1.68/i(^3.25 


(24) 


§32]  RECTANGULAR  COORDINATES  71 

in  which  q  is  the  quantity  of  water  discharged  in  cubic  feet  per 
second,  d  is  the  diameter  of  the  pipe  in  feet  and  h  is  the  loss  of 
head  measured  in  feet  of  water  per  1000  linear  feet  of  pipe. 
This  is  a  good  illustration  of  the  equation  of  a  parabolic  curve 
with  complicated  fractional  exponents.  The  curve  is  very 
roughly  approximate  to  the  locus  of  the  equation 

y=cVhx''''  (25) 

(p)  The  contents  in  gallons  of  a  rectangular  tank  per  foot  of 
depth,  h  feet  wide  and  I  feet  long,  is 

q  =  7.m  (26) 

The  contents  in  gallons  per  foot  of  depth  of  a  cylindrical  tank  d 
feet  in  diameter  is 

g  =  7.5x^2/4  (27) 

Fig.  39  shows  the  graph  of  (26)  for  various  values  of  h  and  also 
shows  to  the  same  scale  the  graph  of  (27). 

32.  Rational  and  Empirical  Formulas.  A  number  of  the 
formulas  given  above  are  capable  of  demonstration  by  means  of 
theoretical  considerations  only.  Such  for  example  are  equations 
(1),  (2),  (3),  (4),  (5),  (7),  (8),  (9),  (10),  etc.,  although  the  constant 
coefficients  in  many  of  these  cases  were  experimentally  deter- 
mined. Formulas  of  this  kind  are  known  in  mathematics  as 
rational  formulas.  On  the  other  hand  certain  of  the  above  for- 
mulas, especially  equations  (14),  (17),  (19),  (22),  (23),  (24), 
including  not  only  the  constant  coefficients  but  also  the  law  oj 
variation  of  the  function  itself,  are  known  to  be  true  only  as  the 
result  of  experiment.  Such  equations  are  called  empirical 
formulas.  Such  formulas  arise  in  the  attempt  to  express  by  an 
equation  the  results  of  a  series  of  laboratory  measurements. 

For  example,  the  density  of  water  (that  is,  the  mass  per  cubic 
centimeter  or  the  weight  per  cubic  foot)  varies  with  the  tem- 
perature of  the  water.  A  large  number  of  experimenters  have 
prepared  accurate  tables  of  the  density  of  water  for  wide  ranges 
of  temperature  centigrade,  and  a  number  of  very  accurate  empirical 
formulas  have  been  ingeniously  devised  to  express  the  results,  of 
which  the  following  four  equations  are  samples: 


72  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§32 

Empirical  formulas  for  the  density,  d,  of  water  in  terms  of  tem- 
perature centigrade,  0. 

m{e  -  4)2 


(a)  d=  I  - 

(b)  d  =  1  - 

(c)  d=l  — 

(d)  d  =  1  + 


107 

93(g  -  4)1-982 

107 

6g2  -  360  +  47 

106 
0.485g3  -  81.302  +  602g  -  1118 
107 

Exercises 


1.  Among  the  power  functions  named  in  the  above  illustrations,  pick 
out  examples  of  increasing  functions  and  of  decreasing  functions. 

2.  Under  the  same  difference  of  head  or  pressure,  show  by  formula 
(24)  that  an  8-inch  pipe  will  transmit  much  more  than  double  the 
quantity  of  water  per  second  that  can  be  transmitted  by  a  4-inch  pipe. 

3.  Wind  velocities  during  exceptionally  heavy  hurricanes  on  the 
Atlantic  coast  are  sometimes  over  140  miles  per  hour.  Show  that  the 
wind  pressure  on  a  flat  surface  during  such  a  storm  is  about  fifty 
times  the  amount  experienced  during  a  20-mile  wind. 

4.  Show  that  for  wind  velocities  of  10,  20,  40,  80,  160  miles  per  hour 
(varying  in  geometrical  progression  with  ratio  2),  the  pressure 
exerted  on  a  flat  surface  is  0.3,  1.2,  4.8,  19.2,  76.8  pounds  per 
square  foot  respectively  (varying  in  geometrical  progression 
with  ratio  4). 

6.  A  300-h.p.  DeLaval  turbine  makes  10,000  revolutions  per  min- 
ute.    Find  the  necessary  diameter  of  the  propeller  shaft. 

6.  A  railroad  switch  target  bent  over  by  the  wind  during  a  tornado 
in  Minnesota  indicated  an  air  pressure  due  to  a  wind  of  600  miles  per 
hour.  Show  that  the  equivalent  pressure  on  a  fiat  surface  would 
be  7.5  pounds  per  square  inch. 

7.  Show  that  a  parachute  50  feet  in  diameter  and  weighing  50 
pounds  will  sustain  a  man  weighing  205  pounds  when  falling  at  the 
rate  of  10  feet  per  second. 

Suggestion:  Use  approximate  value  tt  =  22/7  in  finding  area  of 
parachute  from  formula  for  circle,  itr^'  and  use  formula  (20)   above. 

8.  Show  that  empirical  formulas  (a)  and  (6)  for  the  density  of 
water  reduce  to  a  power  function  if  the  origin  be  taken  at  0  =  4,  d  =  1 . 


§33]  RECTANGULAR  COORDINATES  73 

33.  Practical  Graphs  of  Power  Functions.  The  graphs  of 
the  power  function 

2/  =  rc^,     ij  =  x^,    y  =  l/x,     y  =  x^''\     etc.,  (1) 

can,  of  course,  be  made  the  basis  of  the  laws  concretely  expressed 
by  equations  (l)  to  (27)  of  §31.  If,  however,  the  graph  of  a 
scientific  formula  is  to  serve  as  a  numerical  table  of  the  function 
for  actual  use  in  practical  work,  then  there  is  much  more  labor 
in  the  proper  construction  of  the  graph  than  the  mere  plotting 
of  the  abstract  mathematical  function.  The  size  of  the  unit 
to  be  selected,  the  range  over  which  the  graph  should  extend, 
the  permissible  course  of  the  curve,  become  matters  of  practical 
importance. 

If  the  apparent  slope  ^  of  a  graph  departs  too  widely  from 
+  1  or  —  1,  it  is  desirable  to  make  an  abrupt  change  of  unit  in 
the  A^ertical  or  the  horizontal  scale,  so  as  to  bring  the  curve  back 
to  a  desirable  course,  for  it  is  obvious  that  numerical  readings  can 
best  be  taken  from  a  curve  when  it  crosses  the  rulings  of  the  co- 
ordinate paper  at  apparent  slopes  differing  but  little  from  +  1. 

The  above  suggestions  in  practical  graphing  are  illustrated  by 
the  following  examples: 

Graph  the  formula  (equation  (8),  §31),  for  the  horse  power 
transmissible  by  cold-rolled  shafting 

_  dm  (2) 

in  which  d  is  the  diameter  in  inches  and  N  is  the  number  of 
revolutions  per  minute.  The  formula  is  of  interest  only  for  the 
range  of  d  between  0  and  24  inches,  as  the  dimensions  of  ordinary 
shafting  lie  well  within  these  limits.  Likewise  one  would  not 
ordinarily  be  interested  in  values  of  N  except  those  lying  between 
10  and  3000  revolutions  per  minute.  Fig.  40  shows  a  suitable 
graph  of  this  formula  for  the  range  1  <  d  <  10  for  the  fixed 
value  of  AT  =  100.  In  order  properly  to  graph  this  function,  three 
different  scales  have  been  used  for  the  ordinate  H,  so  that  the 
slope  of  the  curve  may  not  depart  too  widely  from  unity. 

1  Ot  course  the  real  slope  of  a  curve  is  independent  of  the  scales  used.  By 
apparent  slope  =  1  is  meant  that  the  graph  appears  to  cut  the  ruling  of  the 
squared  paper  at  about  45**. 


74 


ELEMENTARY  MATHEMATICAL  ANALYSIS 


§33 


If  similar  graphs  be  drawn  for  N  =  200,  A'  =  300,  A^  =  400, 
etc.,  a  set  of  parabolas  is  obtained  from  which  the  horse  power 
of  shafting  for  various  speeds  of  rotation  as  well  as  for  various 
diameters  may  be  obtained  at  once.  A  set  of  curves  systematically- 
constructed  in  a  manner  similar  to  that  just  described,  is  often 
called  a  family  of  curves.  Fig.  39  shows  a  family  of  straight  lines 
expressing  the  capacity  of  rectangular  tanks  corresponding  to 
the  various  widths  of  the  tanks. 

Inasmuch  as  many  of  the  formulas  of  science  are  used  only  for 

positive  values  of  the  vari- 
ables, it  is  only  necessary  in 
these  cases  to  graph  the 
function  in  the  first  of  the 
four  quadrants.  For  such 
problems  the  origin  may  be 
taken  at  the  lower  left  cor- 
ner of  the  coordinate  paper 
so  that  the  entire  sheet  be- 
comes available  for  the 
curve  in  the  first  quadrant. 
The  above  illustrations 
are  sufficient  to  make  clear 
the  importance  in  science  of 
the  functions  now  being 
discussed.  The  following 
exercises  give  further  prac- 
tice in  the  useful  application  of  the  properties  of  the  functions. 

Exercises 


«'                                                -I 

?•                               tt 

^                               t 

^                                     4- 

§                       i 

«                       t 

!>  en                                               nnr\                                    1 

g50             "     -j4^                t         -, 

|40                      /-400             ^ 

P  on                                   1           nnr                  / 

U                 _,         _L    ^                                  ^ 

0'    1    '    1    1    1    1    '    1    '    '    1    '    '    1    1    '    '    '    '    ' 

123456789      10 
Diameter  of  Shaft  in  Inches 

Fig.    40.— Capacity    at    100    R.P.M.   of 
Cold-rolled  Shafting  to  Transmit  Power. 


The  graphs  for  the  following  problems  are  to  be  constructed  upon 
rectangular  coordinate  paper.  The  instructions  are  for  centimeter 
paper  (form  Ml)  ruled  into  20  X  25  cm.  squares.  In  each  case 
the  units  for  abscissa  and  for  ordinates  are  to  be  so  selected  as  best 
to  exhibit  the  functions,  considering  both  the  workable  range  of 
values  of  the  variables  and  the  suitable  slope  of  the  curves. 

The  student  should  read  §12  a  second  time  before  proceeding 
with  the  following  exercises,  giving  especial  care  to  instructions 
(4),  (5)  and  (6)  given  in  that  section. 


§33]  RECTANGULAR  COORDINATES  75 

1.  Classify  the  graphs  of  formulas  (1)  to  (27),  §31,  as  to 
parabolic  or  hyperbolic  type. 

2.  Graph  the  formula  v'^  =  2gh,  or  y  =  ■\/2gh  =  8.02/i'^-,  if  h  range 
between  1  and  100,  the  second  and  foot  being  the  units  of  measure. 
See  formula  (5),  §31. 

The  following  table  of  values  is  readily  obtained : 

^  i     1         5       10       20^   30 40       50       60       70 80_  90^  ^0 

v\  8.02    17.9   25~3    35.8  43.9    50.7    56.7    62.1    67.1    71.7    76.0    80.2 

Use  2  cm.  =  10  feet  as  the  horizontal  unit  for  h,  and  2  cm.  =  10 
feet  per  second  as  the  vertical  unit  for  v.  The  graph  is  then  readily 
constructed  without  change  of  unit  or  other  special  expedient. 

3.  Graph  the  formula  q  =  3.S7Lh}^  for  L=  1,  for  h=  0,  0.1,  0.2, 
0.3,  0.4,  0.5.  See  formula  (17),  §31.  Use  4  cm.  =  0.1  for  hori- 
zontal unit  for  h  and  2  cm.  =  0.1  for  vertical  unit  for  q. 

4.  Draw  a  curve  showing  the  indicated  horse  power  of  a  ship 
I.H.P.  =  SW^''/C  for  C  =  200  if  the  displacement  D  =  8000  tons,  and 
for  the  range  of  speeds  >S  =  10  to  >S  =  20  knots.  See  formula  (14), 
§31. 

For  the  vertical  unit  use  1  cm.  =  1000  h.p.  and  for  the  horizontal 
unit  use  2  cm.  =  1  knot.  Call  the  lower  left-hand  corner  of  the  paper 
the  point  {S  =  10,  I.H.P.  =  0). 

5.  From  the  formula  expressing  the  centripetal  force  in  pounds  of  a 
rotating  body, 

F  =  0.000341TFi2iV2 

draw  a  curve  showing  the  total  centripetal  force  sustained  by  a  36-inch 
automobile  tire  weighing  25  pounds,  for  all  speeds  from  10  to  40  miles 
per  hour.     See  formula  (13),  §31. 

Miles  per  hour  must  first  be  converted  into  revolutions  per  minute 
by  dividing  5280  by  the  circumference  of  the  tire  and  then  dividing 
the  result  by  60.     This  gives 

1  mile  an  hour  =  9i  revolutions  a  minute  ^ 

If  V  be  the  speed  in  miles  per  hour  the  formula  for  F  becomes 

F  =  0.000341(1.5)25(9^)2  72  =  l.UF^ 

For  horizontal  scale  let  4  cm.  =  10  miles  an  hour  and  for  the  vertical 
scale  let  1  cm.  =  100  pounds. 

6.  Draw  a  curve  from  the  formula  /  =  m/r"^  showing  the  accelera- 
tion of  gravity  due  to  the  earth  at  all  points  between  the  surface  of 


76 


ELEMENTARY  MATHEMATICAL  ANALYSIS 


m 


tlie  earth  and  a  point  240,000  miles  (the  distance  to  the  moon)  from 
the  center,  if  /  =  32.2  when  radius  of  the  earth  =  4000  miles. 

It  is  convenient  in  constructing  this  graph  to  take  the  radius  of 
the  earth  as  unity,  so  that  the  graph  will  then  be  required  of 
/  =  32.2/r2  from  r  =  1  to  r  =  60.  In  order  to  construct  a  suitable 
curve  several  changes  of  units  are  desirable.  See  Fig.  41.  One 
centimeter  represents  one  radius  (4000  miles)  from  r  =  0  to  r  =  10, 
after  which  the  scale  is  reduced  to  1  cm.  =  lOr.  In  the  vertical  direc- 
tion the  scale  is  4  cm.  =  10  feet  per  second  for  0  <  r  <  5,  4  cm.  = 
1  foot  a  second  for  5  <  r  <  10,  and  4  cm.  =0.1  foot  a  second  for 

40 


80 


3:20 
1  - 

>> 

•1     10 


1     2    3    4     5    6     7    8    9  10  20  30  40  50  60 
Distance  from  Eartli's.  Center,  Earth's  Badius>l 

Fig.  41. — Gravitational  Acceleration  at  Various  Distances  from  the 
Earth's  Center.  The  moon  is  distant  approximately  60  earth's  radii  from 
the  center  of  the  earth. 


A_ 

srt 

20 

1 

1 

1 

\ 

I 

\ 

\ 

\ 

^ 

i.\\ 

\ 

> 

V 

^ 

V 

\ 

s 

■v. 

"^ 

\ 

s 

"- 

" 

— 

10  <  r  <  60.  Even  with  these  four  changes  of  units  just  used  the 
first  and  third  curves  are  somewhat  steep.  The  student  can  readily 
improve  on  the  scheme  of  Fig.  41  by  a  better  selection  of  units. 

34.  The  Law  of  the  Power  Functions.  Sufficient  illustrations 
have  been  given  to  show  the  fundamental  character  of  the  power 
function  as  an  expression  of  numerous  laws  of  natural  phenomena. 
How  may  a  functional  dependence  of  this  sort  be  expressed  in 
words?  If  a  series  of  measurements  are  made  in  the  laboratory, 
so  as  to  produce  a  numerical  table  of  data  covering  certain  phe- 


§34]  RECTANGULAR  COORDINATES  77 

nomena,  how  can  it  be  determined  whether  or  not  a  power  function 
can  be  written  down  which  will  express  the  law  (that  is,  the 
function)  defined  by  the  numerical  table  of  laboratory  results? 
The  answers  to  these  questions  are  readily  given.  Consider  first 
the  law  of  the  falling  body 

s  =  16.1^2  (1) 

Make  a  table  of  values  for  values  of  i  =  1,  2,  4,  8,  16  seconds,  as 
follows: 


16 


16.1    64.4     257.6    1030.4    4121.6 

The  values  of  t  have  been  so  selected  that  t  increases  by  a  fixed 
multiple;  that  is,  each  value  of  t  in  the  sequence  is  twice  the  pre- 
ceding value.  From  the  corresponding  values  of  s  it  is  observed 
that  s  also  increases  by  a  fixed  multiple,  namely  4. 

Similar  conclusions  obviously  hold  for  any  power  function. 
Take  the  general  case 

y  =  ax»  (2) 

where  n  is  any  exponent,  positive,  negative,  integral  or  fractional. 
Let  X  change  from  any  value  Xi  to  a  multiple  value  mxi  and  call 
the  corresponding  values  of  y,  yi  and  2/2.     Then  we  have 

2/1  =  «^i"  (3) 

and 

2/2  =  a(mxi)''  =  «?7i»a;i«  (4) 

Divide  the  members  of  (4)  by  the  members  of  (3)  and  we  have 

;-:=.»  (5) 

That  is,  if  x  in  any  power  function  change  by  the  fixed  multiple 
m,  then  the  value  of  y  will  change  by  a  fixed  multiple  m".  Thus 
the  law  of  the  power  function  may  be  stated  in  words  in  either  of 
the  two  following  forms: 

In  any  power  function,  if  x  change  by  a  fixed  multiple,  y  will 
change  by  a  fixed  multiple  also. 

In  any  power  function,  if  the  variable  increase  by  a  fixed  percent, 
the  function  will  increase  by  a  fixed  percent  also. 


78  ELEMENTARY   MATHEMATICAL  ANALYSIS        I  §35 

This  test  may  readily  be  applied  to  laboratory  data  to  determine 
whether  or  not  a  power  function  can  be  set  up  to  represent  as  a 
formula  the  data  in  hand.  To  apply  this  test,  select  at  several 
places  in  one  column  of  the  laboratory  data,  pairs  of  numbers 
which  change  by  a  selected  fixed  percent,  say  10  percent,  or  20 
percent,  or  any  convenient  percent.  Then  the  corresponding  pairs 
of  numbers  in  the  other  column  of  the  table  must  also  be  related  by 
a  fixed  percent  (of  course,  not  in  general  the  same  as  the  first- 
named  percent),  provided  the  functional  relation  is  expressible  by 
means  of  a  power  function.  If  this  test  does  not  succeed,  then 
the  function  in  hand  is  not  a  power  function. 

Since  the  fixed  percent  for  the  function  is  w"  if  the  fixed  percent 
for  the  variable  be  m,  the  possibility  of  determining  n  exists, 
since  the  table  of  laboratory  data  must  yield  the  numerical  values 
of  both  m  and  m". 

35.  Simple  Modifications  of  the  Parabolic  and  of  the  Hyperbolic 
Types  of  Curves.  In  the  study  of  the  motion  of  objects  it  is 
convenient  to  divide  bodies  into  two  classes:  first,  bodies  which 
retain  their  size  and  shape  unaltered  during  the  motion;  second, 
bodies  which  suffer  change  of  size  or  shape  or  both  during  the 
motion.  The  first  class  of  bodies  are  called  rigid  bodies;  a  mov- 
ing stone,  the  reciprocating  or  rotating  parts  of  a  machine,  are 
illustrations.  The  second  class  of  bodies  are  called  elastic  bodies ; 
a  piece  of  rubber  during  stretching,  a  spring  during  elongation  or 
contraction,  a  rope  or  wire  while  being  coiled,  the  water  flowing  in 
a  set  of  pipes,  are  all  illustrations  of  this  class  of  bodies. 

When  a  body  changes  size  or  shape  the  motion  is  called  a 
strain. 

Bodies  that  preserve  their  size  and  shape  unchanged  may  possess 
motion  of  two  simple  types:  (1)  Rotation,  in  which  all  particles 
of  the  body  move  in  circles  whose  centers  lie  in  a  straight  line 
called  the  axis  of  rotation,  which  line  is  perpendicular  to  the  plane 
of  the  circles,  and  (2)  translation,  in  which  each  straight  line  of 
the  body  remains  fixed  in  direction. 

We  have  already  noted  that  the  curve 


2/1  _ 


X-  (1) 


§36]  RECTANGULAR  COORDINATES  79 

can  be  made  from  the  curve 

y  =a:"  (2) 

by  multiplying  all  the  ordinates  of  (2)  by  a.  The  effect  is  either  to 
elongate  or  to  contract  all  of  the  ordinates,  depending  upon  whether 
a  >  1  or  a  <  1  respectively.  The  substitution  of  (?/i  ja)  for  y 
has  therefore  produced  a  motion  or  strain  in  the  curve  y  =  x",  which 
in  this  case  is  the  object  whose  motion  is  being  studied.    Likewise 

y=(xilay  (3) 

can  be  made  from 

y  =  x^  (4) 

by  multiplying  all  of  the  abscissas  of  (4)  by  a.     The  effect  is 
either  to  stretch  or  to  contract  all  of  the  abscissas,  depending 
upon  whether  a  >  1,  or  a  <  1  respectively. 
In  general,  if  a  curve  have  the  equation 

y  =  m  (5) 

then 

y=Kx,la)  (6) 

i  made  from  curve  (5)  by  lengthening  or  stretching  the  XY- 
plane  uniformly  in  the  x  direction  in  the  ratio  1 :  a. 

The  statement  just  given  is  made  on  the  assumption  that 
a  >  1.  If  a  <  1  then  the  above  statements  must  be  changed 
by  substituting  shorten  or  contract  for  elongate  or  stretch. 

The  reasons  for  the  above  conclusions  have  been  previously 

stated:  substituting  (  — j  everywhere  as  the  equal  of  x  multiplies 

all  of  the  abscissas  by  a.     That  is,  if  (  - )  =  x,  thenxi  =  ax,  so  that 

Xi  is  a-fold  the  old  x. 

We  shall  now  explain  how  certain  other  of  the  motions  men- 
tioned above  may  be  given  to  a  locus  by  suitable  substitution  for 
x  and  y. 

36.  Translation  of  Any  Locus.  If  a  table  of  values  be  prepared 
for  each  of  the  loci 

y  =  x^  (1) 


80  ELEMENTARY  MATHEMATICAL  ANALYSIS         [§36 

as  follows: 

xl     -2     -101234 


y  \         4  10149    16 

a;i|-2       -10123456 
y  \        25         16    9410149 

and  then  if  the  graph  of  each  be  drawn,  it  will  be  seen  that  the 
curves  differ  only  in  their  location  and  not  at  all  in  shape  or  size. 
The  reason  for  this  is  obvious:  if  (o^i  —  3)  be  substituted  for  x 
in  any  equation,  then  since  {xi  —  3)  has  been  put  equal  to  x,  it 
follows  that  xx  =  X  -\-  Z,  or  the  new  x,  namely  Xi,  is  greater 
than  the  original  x  by  the  amount  3.  This  means  that  the  new 
longitude  of  each  point  of  the  locus  after  the  substitution  is  greater 
than  the  old  longitude  by  the  fixed  amount  3.  Therefore  the 
new  locus  is  the  same  as  the  original  locus  translated  to  the  right 
the  distance  3. 

The  same  reasoning  applies  if  {xi  —  a)  be  substituted  for  x, 
and  the  amount  of  translation  in  this  case  is  a.  The  same  reason- 
ing applies  also  to  the  general  case  y  =  f{x)  and  y  =  f(xi—  a), 
the  latter  curve  being  the  same  as  the  former,  translated  the  dis- 
tance a  in  the  x  direction. 

As  it  is  always  easy  to  distinguish  from  the  context  the  new  x 
from  the  old  x,  it  is  not  necessary  to  use  the  symbol  xi,  since  the 
old  and  new  abscissas  may  both  be  represented  by  x.  The 
following  theorems  may  then  be  stated: 

Theoeems  on  Loci 

VI.  If  (x  —  a)  be  substituted  for  x  throughout  any  equation,  the 
locus  is  translated  a  distance  a  in  the  x  direction. 

VII.  If  {y  —  b)  be  substituted  for  y  in  any  equation,  the  locus  is 
translated  the  distance  b  in  the  y  direction. 

These  statements  are  perfectly  general:  if  the  signs  of  a  and 
b  are  negative,  so  that  the  substitutions  for  x  and  y  are  of  the  form 
x  +  a'  and  y  +  b',  respectively,  then  the  translations  are  to 
the  left  and  down  instead  of  to  the  right  and  up. 

Sometimes  the  motion  of  translation  may  seem  to  be  disguised 
by  the  position  of  the  terms  a  or  b.     Thus  the  locus  y  =  Sx  +  5 


§36]  RECTANGULAR  COORDINATES  81 

is  the  same  as  the  locus  y  =  3x  translated  upward  the  distance  5, 
for  the  first  equation  is  really  y  —  5  =  Sx,  from  which  the  conclu- 
sion is  obvious. 

Exercises 

1.  Compare  the  curves:  (1)  y  =  2x  and  y  =  2(a;  —  1);  (2)  y  =  x^ 
and  y  =  {x  —  4)^;  (3)  y  =  x^  and  y  —  S  =  x^]  (4)  y  =  x^  and 
y  =  {x  -  5)?^;  (5)  y  =  5x^  and  y  =  5{x  +  3)^;  (6)  y  =  2x^  and 
y  =  2ix  -  ky;  (7)  y  =  2x^  and  y  =  2x^  +  k;  (8)  y  +  7  =  x^  and 
y  =  x^  and  y  -  7  =  x^;  (9)  3i/2  =  Sx^  and  d{y  -  6)2  =  5{x  -  a) 3. 

2.  Compare  the  curves:  (1)  y  =  x^  and  y  =  (x/2)^;  (2)  y  =  x^ 
and  2/  =a;V8;  (3)  ?/  =  a;^  and  y/2  =  x^\  (4)  y  =  x^  and  t/  =  2a;3;  (5) 
2/2  =  3x3  and  (7//5)2  =  3(x/7)3;  (6)  t/^  =  ^''  and  2/2  =  (3x)3;  (7) 
1/  =  a;2  and  ?/  =  4x2  (note:  explain  in  two  ways);  (8)  y  =  x^  and 
2y  =  x^  and  ?/  =  27a;3. 

3.  Translate  the  locus  y  =  2x^\  (1)  3  units  to  the  right;  (2)  4  units 
down;  (3)  5  units  to  the  left. 

4.  Elongate  three-fold  in  the  x  direction  the  loci:  (1)  y'^  =  x;  (2) 
3y  =  x^;  (3)  y^  =  2x';  (4)  ?/  =  2x  +  7. 

5.  Multiply  by  1/2  the  ordinates  of  the  loci  named  in  exercise  4. 

6.  Show  that  y  =  ,   and  y  =  7  are  hyperbolas. 

•C     ]~  0  X  0 

X 


7.  Show  that  y  =  — -—f  is  a  hyperbola 

3J  "f"  0 


Note:    Divide  the  numerator  by  the  denominator,  obtaining  the 

b 

equations  =  1 r^- 

x  +h 

8.  Show  that  y  =  — — -r  is  a  hyperbola,  namely,  the  curve  xy  =  a—b 
X  ~\~  0 

translated  to  a  new  position. 

37.  Shearing  Motion.    An  important  strain  of  the  XF-plane 
occurs  if  we  derive 

y  =  fix)  +  wa;  (1) 

from 

y'  =  fix)  (2) 

Graphically,  the  curve  (1)  is  seen  to  be  formed  by  the  addition 
of  the  ordinates  of  the  straight  line  y"  =  mx  to  the  corresponding 
ordinates  of  y'  =  f{x).     Thus,  in  Fig.  42,  the  graph  of  the  func- 

6 


82 


ELEMENTARY  MATHEMATICAL  ANALYSIS        [§37 


tion  x^  -\-  X  is  made  by  adding  the  corresponding  ordinates  of 
y'  =^  x^  and  y"  =  x.  Mechanically,  this  might  be  done  by  drawing 
the  curve  on  the  edge  of  a  pack  of  cards,  and  then  slipping  the 
cards  over  each  other  uniform  amounts.  The  change  of  the 
shape  of  a  body,  or  the  strain  of  a  body,  here  illustrated,  is 
called  lamellar  motion  or  shearing  motion.  It  is  a  form  of 
motion  of  very  great  importance. 


4 
3 

l/l/  ■' 
/  /    ^ 

\ 

9. 

/ 

\ 

1        / 

ll 

\ 

/ 

-3 

~^ 

I 

¥ 

-1       \ 

A 

I           3 

4 

-9. 

\ 

/ 

-3 

\ 

-4 

Fig.  42. — The  Shear  of  the  Cubical  Parabola  y  =  x^  in  the  line  y  =  x,  and 
also  in  the  Line  y    =  —x. 

We  shall  speak  of  the  locus  y  =  f(x)  +  mx  as  the  shear  of  the 
curve  y  =  f{x)  in  the  line  y  =  mx. 


Theorems  on  Loci 


VIII.  The  addition  of  the  term  mx  to  the  right  side  of  y  =   f{x) 
shears  the  locus  y=f{x)  in  the  line  y  =  7nx. 
The  locus 

y  =  ax^  -^  mx  -j-  h 


§37J 


RECTANGULAR  COORDINATES 


83 


is  made  from  y  =  x^  by  a  combination  of  (1)  a  uniform  elongation 
a],  (2)  a  shearing  motion  [m],  and  (3)  a  translation  [6].  Either 
motion  may  be  changed  in  sense  by  changing  the  sign  of  a,  w, 
or  h,  respectively. 

The  student  may  easily  show  that  the  effect  of  a  shearing  motion 
upon  the  straight  line  y  =  mx  -{-  b  is  merely  a  rotation  about 
the  fixed  point  (0,  6).     The  line  is  really  stretched  in  the  direction 


H 

K 

t  —  -.l"o^ 

8      "'      "" 

7- 

6- 
5- 
4-                          0 

0 

1-             o 

0 
0 

o 
o 

o 

1 
1  - 

2- 

3- 

4- 

5- 

6- 

7  - 

I 

H 


In^    K 


Fig.  43. 


-Shearing  Motion  Illustrated  by  the  Slipping  of  the  Members  of 
Pack  of  Cards. 


of  its  own  length,  but  this  does  not  change  the  shape  of  the  line 
nor  does  it  change  the  line  geometrically.  A  line  segment  (that 
is,  a  line  of  finite  length)  would  be  modified,  however. 

The  parabola  y  =  x^  is  transformed  under  a  shearing  motion 
in  a  most  interesting  way.     For,  after  shear,  y  =  .t'  becomes: 


y  =  :r2  +  2mx 


(3) 


84 


ELEMENTARY  MATHEMATICAL  ANALYSIS 


:§37 


where,  for  convenience,  the  amount  of  the  shearing  motion  is 
represented  by  2m  instead  of  by  m.     Writing  this  in  the  form 


or, 


=  x^  -\-  2mx  +  m^  - 

y  =  {x  -\-  my  —  m 
y  -{-  m^  =  (x  -\-  m)' 


(4) 


we  see  that  (4)  can  be  made  from  the  parabola  y  =  x^  by  trans- 
lating the  curve  to  the  left 
the  amount  m  and  down 
the  amount  m^.  (See  Fig. 
44.) 

Shearing  motion,  there- 
fore, rotates  the  straight 
line  and  translates  the  pa- 
rabola. The  effect  on  other 
curves  is  much  more  com- 
plicated, as  is  seen  from 
Figs.  42  and  43. 

The  parabola  y  =  a:^  is 
identical  in  size  and  shape 
with  y  =  x^  -{-  mx  -\-  b. 
Likewise,  y  =  ax^  -}-  bx  -\-  c 
is  a  parabola  differing  only 
in  position  from  y  =  ax-. 

Exercises 


\ 

\ 

4 
3 

1 

\ 

\\ 

9. 

•  n 

\\ 

,     / 

1    .^ 

5^ 

\ 

L 

/ 

/ 

-3 

-2 

r 

0 

-1 

5 

I            3 

^ 

y^ 

-5» 

-S 

-4         . 

FiQ.  44. 


-The  Shear  of  y 
y  =  0.6a;. 


in  the  line 


1  Explain  how  the  curve 
y  =  x^  +  2x  may  .be  made 
from  the  curve  y  =  x^.     How  can  the  curve  y  =  2x^  +  dx  be  made 
from  the  curve  y  =  2a;  ^? 

2.  Find   the    coordinates   of   the   lowest   point   of   y  =  x^  —  4.x; 
that  is,  put  this  equation  in  the  form  y  —  b  =  {x  —  a)^, 

3.  Compare  the  curves  y  =  x^  +  2x  and  y  =  x^  —  2x.     (Do  not 
draw  the  curves.) 

4.  Explain  the  curve  y  =  1/a;  +  2a;  from  a  knowledge  oi  y  =  1/x 
and  oi  y  =  2x. 


§39]  RECTANGULAR  COORDINATES  85 

38.  Rotation  of  a  Locus.  The  only  simple  type  of  displace- 
ment of  a  locus  not  yet  considered  is  the  rotation  of  the  locus 
about  the  origin  0.  This  will  be  taken  up  in  the  next  chapter 
in  the  discussion  of  a  new  system  of  coordinates  known  as  polar 
coordinates.  The  rotation  of  any  locus  about  the  X-axis  or  about 
the  F-axis  is  readily  accomplished,  however,  as  previously  ex- 
plained. For  substituting  ( —  x)  for  x  changes  every  point  that  is 
to  the  right  of  the  F-axis  to  a  point  to  the  left  thereof,  and  vice 
versa.  It  is  equivalent,  therefore,  to  a  rotation  of  the  locus  about 
the  F-axis.  Likewise,  substituting  {—  y)  for  y  rotates  any  locus 
180°  about  the  X-axis.  It  is  preferable,  however,  to  speak  of  the 
locus  formed  in  this  way  as  the  reflection  of  the  original  curve  in 
the  i/-axis  or  in  the  x-axis,  as  the  case  may  be. 

39.  Roots  of  Functions.  The  roots  or  zeros  of  a  function  are 
the  values  of  the  argument  for  which  the  corresponding  value  of 
the  function  is  zero.  Thus,  2  and  3  are  roots  of  the  function ' 
a; 2  —  5a;  -f-  6,  for  substituting  either  number  for  x  causes  the 
function  to  be  zero.  The  roots  of  x^  —  a;  —  6  are  +  3  and  —  2. 
The  roots  of  x^  -  ^x'^ -\- llx  -  ^  are  1,  2,  3. 

The  word  root,  used  in  this  sense,  has,  of  course,  an  entirely 
different  significance  from  the  same  word  in  "square  root,"  "cube 
root,"  etc.  But  the  roots  of  the  function  x^  —  5a;  *-  6  are  also  the 
roots  of  the  equation  x*  —  5a;  —  6  =  0. 

In  the  graph  of  the  cubic  function  y  =  x^  —  x  in  Fig.  42,  the 
curve  crosses  the  X-axis  at  a;  =  —  1,  a;  =0,  and  a;  =  1.  These  are 
the  values  of  x  that  make  the  function  a;'  —  a;  zero,  and  are,  of 
course,  the  roots  of  the  function  x^  —  x.  No  matter  what  the  func- 
tion may  be,  it  is  obvious  that  the  intercepts  on  the  X-axis,  as  OA, 
OB,  Fig.  42,  must  represent  the  roots  of  the  function. 

Exercises 

1.  From  the  curve  y  =  x^  sketch  the  curves  y  —  4  =  x^;  y  =  4x^; 
4y  =  x^;  y  =  {x  —  4)2. 

(x  —  3^)* 

2.  Sketch  y  =  xy2;  y  =  x^  -  1/4;  y  =  xy2  -  4;  y  =^— --^-. 

3.  Sketch  the  curves  y  =  \/x;  y  =  Vx;  y=  2Vx;  y  =  Vx^2j 
y  -  2  =  Vx  -2,  and  y  =  ^x  -  3. 


86 


ELEMENTARY  MATHEMATICAL  ANALYSIS        [§40 


4.  Sketch  the  curves  y^  =  (x  -  Sy;  (y  -  2^  =  x\  and  {y  -  2y 

=  (x  -  sy. 

5.  Graph  ?/i  =  x  and  7/2  =  x^  and  thence  y  =  x  -{-  x^. 

6.  Find    the    roots    of    x^  —  6.r  +  8  =  0,    from    the    graph    of 
y  =  x^  -  Qx  +  8. 


7.  Find  the  roots  of  the  functions  x^  —  a^  and  x^  —  a^. 

8.  Compare  the  curves  y  =  x^  and  y  =  —  x^;y  =  x^  and 


y 


^  =  2x  +  3  and  2/  =  -  2x  +  3. 

9.  Graph  yi  =  x  and  2/2  =  l/x  and  thence  y  =  x  +  1/x. 

10.  Compare  y  =  l/x,  y  =  l/{x  -  2),  y  =  l/(x  +  3). 

11.  Compare  y  =  l/x,  y  =  1/(23:),  ?/  =  2/x. 

40.  *Graphical  Construction  of  Power  Functions  and  of  other 
Functions.^     The   graphical  computation  of   products   and  quo- 


Y 

V^ 

/ 

R 

\ 

A 

A- 

\ 

4 

1 

\ 

\ 

1 

1.1 

A 

\ 

U, 

\ 

/ 

b 

\ 

/ 

/  y 

m' 

a 

> 

cd 

\ 

/y 

' 

a 

a 

\ 

J 

1 

ac 

/ 

\° 

D 

; 

■/ 

\ 

c 

x' 

-\- 

d 

y 

/ 

\ 

V 

/> 

/ 

/ 

\ 

' 

/ 

\ 

— 

~ 

— 

y 

/ 

\ 

/ 

Y 

\ 

Fig.  45. — Construction  of  an  Ordinate  Equal  to  the  Product   of   Two 
Given  Ordinates, 

tients,  etc.,  explained  in  §7,  may  be  applied  to  the  construction  of 
the  power  functions.  For  this  purpose  it  is  desirable  to  elaborate 
slightly  the  previous  method  so  as  to  provide  for  finding  prod- 
ucts, etc.,  of  lines  that  are  parallel  to  each  other,  instead  of  at  right 
angles  as  OA  and  IB,  Fig.  9. 

1  The  remainder  of  this  chapter  (except  the  review  exercises)  may  be  omitted  with- 
out loss  of  continuity. 


RECTANGULAR  COORDINATES 


87 


The  constructions  can  be  carried  out  on  plain  paper  by  first 
drawing  the  axes,  the  unit  lines  and  the  line  y  =  x,  without  the 
use  of  scales  or  measuring  device  of  any  sort.  The  work  is  more 
rapidly  done,  however,  on  squared  paper,  as  then  the  use  of  a 
T-square  and  triangle  may  be  dispensed  with.  A  unit  of  measure 
equal  to  2  inches  or  4  cm.  will  be  found  convenient  for  work 
on  standard  letter  paper  8^  X  11  inches. 

Note  that  the  following  constructions  give  both  the  magnitude 
and  the  proper  algebraic  sense  of  the  results. 

(1)  To   construct   an    ordinate  , , 

eqibal  to  the  product  of  two  ordi- 
nates:  Let  XX',  YY',  Fig.  45,  b  e 
the  axes,  Ui,  U2  the  unit  Hnes, 
and  OR   the  line  y  =  x,  which 


Fig.  46. — Construction  of  an  Or- 
dinate Equal  to  the  Quotient  of 
Two  Given  Ordinates. 


Fig.  47. — Construction  of  an  Or- 
dinate Equal  to  the  Square  of  a 
Given  Ordinate. 


we  shall  call  the  reflector.  Let  a  and  b  be  two  ordinates  whose 
product  is  required.  Move  one  of  the  two  given  ordinates  as  b 
until,  in  the  position  ND,  its  end  touches  the  reflector  OR. 
Move  the  second  of  the  two  ordinates  to  the  position  IM  on  the 
unit  line  Ui.  Draw  OMP.  The  point  P  at  which  DAT  is  cut  by 
OM  (produced  if  necessary)  determines  DP,  which  is  the  prod- 
uct aXb.  This  result  follows  by  similar  triangles  from  the  pro- 
portion 

DP-AM  =  OD:Ol 


88 


ELEMENTARY  MATHEMATICAL  ANALYSLS 


[§40 


Substituting  a  for  IM  and  h  for  OD  (=  DN  =  h)  and  unity  for 
01,  we  obtain 

DP:a  =  h\l 
or 

DP  =  aXb 

The  same  diagram  shows  the  construction  of  the  products  c  X  d 
and  a  X  c  for  cases  in  which  one  or  both  of  the  factors  are  negative. 

Note  that  by  the  above  construction  the  ordinate  representing 
the  product  is  always  located  at  a  particular  place,  D,  at  which  the 
abscissa  of  the  product  aXbis  either  equal  to  a  or  to  b,  depending 
upon  which  of  the  ordinates  was  moved  to  the  reflector  OR. 


Y 

Ui 

A       U: 

^^^' 

^-^'        1 

'-          -jji  ^ 

.1 

X            ^ 

Fig.  48. — Construction  of  the  Reciprocal  of  x. 


(2)  To  construct  an  ordinate  equal  to  the  quotient  of  two  ordi- 
nates: This  is  done  by  use  of  the  second  unit  line  U^  as  shown  in 
Fig.  46.  The  ordinate  representing  the  quotient  is  located  at  D 
where  OD  equals  the  dividend  b. 

(3)  The  special  case  of  (1)  when  a  =  b  leads  to  the  construction 
of  x2  as  shown  in  Fig.  47.  The  figure  shows  the  construction  of 
x^  at  D  where  OD  =  x  and  of  Xi^  at  Di  where  ODi  =  Xi. 

(4)  The  special  case  of  (2)  where  6=1  leads  to  the  construc- 
tion of  1/x  as  shown  in  Fig.  48. 

(5)  To  construct  the  graph  of  y  =  x^,  it  is  merely  necessary  to 
make  repeated  appHcations  of  (3)  to  the  successive  ordinates  of 


m 


RECTANGULAR  COORDINATES 


89 


the  line  y  =  x,  as  shown  in  Fig.  49.  Thus  from  any  point  A  of 
y  =  X  move  horizontally  to  the  unit  line  Ui  locating  B,  then  if 
OB  meets  DA  at  P,  P  is  a  point  of  the  curve  y  =  x^.  The  figure 
shows  the  construction  for  a  number  of  points,  lettered  Ai,  A2, 
A3,  .     .     . 

(6)  To  construct  the  graph  of  y  =  x^,  first  cut  out  a  pattern  of 
y  =  x"^  oi  heavy  paper,  marking  upon  it  the  lines  OY  and  U2 


Y 

U^ 

\ 

fr 

— 

- 

V 

A 

h 

V 

s 

/ 

/ 

\ 

L^ 

B^ 

// 

/ 

V 

\, 

/ 

p 

A 

\ 

N 

0.] 

A 

r 

^ 

^^ 

>2 

B. 

V, 

N 

^ 

\ 

/ 

^^ 

-P2 

— 

Pg 

^ 

^ 

^ 

^ 

X 

^ 

% 

^ 

--'' 

^. 

% 

~-.D2 

1.0 

Dr 

X 

/ 

s^  V, 

/ 

>>\ 

S3 

/ 

A 

J 

W 

/ 

\\ 

/ 

\ 

/ 

/ 

L 

- 

- 

— 

— 

— 

J 

— 

Y 

J 

— 

Fig.  49. — Construction    of   the    Curve   y  =  x^   from   the   curve  y  =  x. 


By  means  of  this  pattern  draw  the  curve  y  =  x^  upon  a  fresh  sheet 
of  paper  as  shown  in  Fig.  50.  Then  multiply  each  ordinate  of 
y  =  x^hy  xhy  moving  it  horizontally  from  any  point  A  oiy  =  x^ 
to  the  unit  line  C7i  at  P,  then  locating  P  on  DA  by  drawing  OB 
until  it  cuts  DA  at  P.     The   result  is   the   cubical   parabola 

PlP2P4P'4P'2. 

(7)  To  draw  the  hyperbola  y  =  1/x,  make  repeated  application 
of  (4)  above  to  successive  values  of  x.  To  draw  y=  l/x"^,  repeat 
division  by  x  to  the  ordinates  of  y  =  1/x,  etc. 


90 


ELEMENTARY  MATHEMATICAL  ANALYSIS 


}40 


(8)  To  construct  the  graph  of  y  =  x"'^'.  First,  from  the  pattern 
of  2/  =  a;  2  draw  the  curve  y  =  \/  x.  From  a  pattern  draw  the 
curve  y  —  x^  upon  the  same  axes.  Then  from  any  point  Ax 
oi  y  =  x''^  proceed  horizontally  to  Bi  on  the  reflector;  then  ver- 
tically to  Ci  on  the  curve  y  =  x^,  then  horizontally  to  Pi  on  the 
ordinate  DAi  first  taken.  Then  Pi  is  on  the  curve  y  =  x^^\ 
For,  call  DAi  =  y^;  HCx  =  2/2;  DPx  =  y;  OD  =  x;  OH  =  x-2. 
Then  by  construction  (Fig.  51) 


\                  ^         ^'      I 

A                                 ^m^ 

\-                    zx 

t                    -yLJI 

A                       ttt 

i                            :?/l/^ 

t                 7N7     ^ 

X                 t/l   / 

i              jBUWi 

\         Lti/ 

i             -U  t 

\            0.1//  4\,\              £/, 

^^      -U-^t 

ipS 

X                               /'^^O.O              l.ODaO            X 

JpM- 

l¥-Q- 

^/A 

^    '!^ 

^t^l 

^  u~t 

^  Jp.t 

Ul 

Fig.  50. — Construction  of  the   Curve  y  =  x^  from   the   curve  y  =  0-2. 

OH  =  DA  =yx  =  x^''  (1) 

DPx  =  y  =  HCy  =  2/2  =  X2'  (2) 


But, 

Hence,  by  (3)  and  (2): 

and  by  (1) 


y  =  yi"^ 
y  =  {^"Y  =  ^  ~ 


(3) 


(5) 


}40] 


RECTANGULAR  COORDINATES 


91 


(9)  Function  of  a  function:  The  construction  and  reasoning 
just  given  applies  to  a  much  more  general  case.  Thus  if  the  curve 
OAi,  Fig.  51,  has  the  equation 

y  =  fix) 

and  if  the  curve  OCi  has  the  equation 

y  =  F{x) 


Y 

u 

/ 

/ 

/ 

/ 

A 

?? 

/ 

/ 

/ 

/ 

- 

— 

/ 

^' 

''' 

1 

,y 

IJ 

u. 

A 

y/ 

7] 

B 

Pi 

/ 

/, 

A 

0 

^ 

^c 

'l 

X 

^ 

^"'^ 

H 

1 

X 

/ 

f/f 

/ 

s 

2 

/ 

Sf 

/ 

... 

^N. 

S 

1/ 

Bi 

/ 

12^ 

^ 

A 

\" 

/ 

/  \ 

s 

**.. 

/ 

\ 

/ 

^ 

/ 

1 

y{ 

Fig.  51. — Construction  of  the  Semi-cubical  Parabola  y  =  x^i  from  y 
X'  and  y  =  xH 


then  the  curve  OPi  has  the  equation 

y  =  FUix)] 

Thus,  if  OA I  be  the  curve 

y  =  1  —  x^ 
and  OCi  the  curve 


then  OP  I  is  the  curve 


y  =  X-'- 


2/  =  (1  -  x'^^ 


•5^ 


92  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§40 

For  constructions  of  the  function 

y  =  a  -{-  aix  -{-  a^x^  +  .    .    .  +  a„a;« 
see  "Graphical  Methods"  by  Carl  Runge,  Columbia  University 
Press,  1912. 

Miscellaneous  Exercises 

1.  Define  a  function.  Explain  what  is  meant  by  a  discontinuous 
function.     Give  practical  illustrations. 

2.  Define  an  algebraic  function;  rational  function;  fractional 
function.     Give  practical  illustrations  in  each  case. 

3.  Give  an  illustration  of  a  rational  integral  function;  of  a 
rational  fractional  function. 

4.  Write  a  short  discussion  of  the  Cartesian  method  of  locating  a 
point.     Explain  what  is  meant  by  such  terms  as  "axis,"  "x  of  a 
point,"  "quadrant,"  etc. 

5.  What  is  meant  by  the  locus  of  an  equation? 

6.  Write  the  equations  of  the  lines  determined  by  the  following 
data :  • 

(a)  slope      2  F-intercept  5 

(6)  slope  —2  F-intercept  5 

(c)  slope      2  F-intercept  —5 

{d)  slope  —2  F-intercept  —5 

(c)  slope  —2  X-intercept  4 

7.  Make  two  suitable  graphs  upon  a  single  sheet  of  squared  paper 
from  the  following  data  giving  the  highest  and  lowest  average  clos- 
ing price  of  twenty-five  leading  stocks  listed  on  the  New  York  Stock 
Exchange  for  the  years  given  in  the  table : 

Year  Highest  Lowest 

1913  94.56  79.58 

1912  101.40  91.41 

1911  101.76  86.29 

1910  111.12  86.32 

1909  112.76  93.24 

1908  99.04  67.87 

1907  109.88  65.04 

1906  113.82  93.36 

1905  109.05  90.87 

1904  97.73  70.66 

1903  98.16  68.41 

1902  101.88  87.30 


§40]  RECTANGULAR  COORDINATES  93 

Should  smooth  curves  be  drawn  through  the  points  plotted  from  this 
table? 

8.  Define  a  parabolic  curve.    What  is  the  equation  of  the  parabola  ? 
Of  the  cubical  parabola?     Of  the  semi-cubical  parabola?     , 

9.  What  is  the  definition  of  an  hyperbolic  curve?    Of  the  rectangu- 
lar hyperbola? 

10.  Draw  on  a  sheet  of  coordinate  paper  the  lines  a;  =  0,  x  =  1, 
X  =  —1,  2/  =  0,  2/  =  1,  y  =  —1.  Shade  the  regions  in  which  the 
hyperbolic  curves  lie  with  vertical  strokes;  and  those  in  which  the 
parabolic  curves  lie  with  horizontal  strokes.  Write  down  all  that  the 
resulting  figure  tells  you. 

11.  Consider  the  following :  y  =  x^,  y  =  x~^,  y  =  ^x^,  xy  =  —1, 
y  =  -x^,  y^  =  x\  y^  =  x^,  xy  =  1,  x^  =  -  2/^  x*  =  -  y^.  Which 
are  increasing  functions  of  x  in  the  first  quadrant?*  For  which 
does  the  slope  of  the  curve  increase  in  the  first  quadrant?  For 
which  does  the  slope  of  the  curve  decrease  in  the  first  quadrant? 

12.  Which  of  the  curves  of  exercise  11  pass  through  (0,  0*)? 
Through  (1,  1)?    Through  (-1,  -1)? 

13.  Find  the  vertex  of  the  curve  y  =  x^  —  24x  +  150. 

Note:  The  lowest  point  of  the  parabola  y  =  x^  may  be  called 
the  vertex. 

Suggestion:  It  is  necessary  to  put  the  equation  in  the  form  y  —  b 
=  (x  —  ay.  This  can  be  done  as  follows :  Add  and  subtract  144  on 
the  right  side  of  the  equation,  obtaining 

y  =x^  -2ix  +  144  -  144  +  150 
or, 

y   ={x-  12)2  +  6 
or, 

y-Q  =  {X-  12)2 

Then  this  is  the  curve  y  =  x^  translated  12  units  to  the  right  and  6 
units  up.  Since  the  vertex  oi  y  =  x^  is  at  the  origin,  the  vertex  of  the 
given  curve  must  be  at  the  point  (12,  6). 

14.  Find  the  vertex  of  the  parabola  y  =  x"^  —  Qx  +11. 
16.  Find  the  vertex  oi  y  =  x^  +  Sx  +  1. 

16.  Find  the  vertex  oi  4.  +  y  =  x^  -  7x. 

17.  Find  the  vertex  oiy=  9x^  +  18a;  +  1. 

18.  Translate  y  =  4:X^  —  12a;  +  2  so  that  the  equation  may  have 
the  form  y  =  4x2. 


CHAPTER  III 

THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS 

41.  Equation  of  the  Circle.  In  rectangular  coordinates  the 
abscissa  x,  and  the  ordinate  y,  of  any  point  P  (as  OD  and  DP, 
Fig.  52)  form  two  sides  of  a  right  triangle  whose  hypotenuse 
squared  is  x^  +  ?/2.  If  the  point  P  move  in  such  manner  that  the 
length  of  this  hypotenuse  remains  fixed,  the  point  P  describes  a 
circle  whose  center  is  the  origin  (see  Fig.  52).     The  equation  of 

this  circle  is,  therefore: 

Y 


x2  +  y2  =  q} 


(1) 


if  a  stand  for  OP,  Fig.  52,  namely 
the  fixed  length  of  the  hypote- 
nuse, or  the  radius  of  the  circle. 
It  is  sometimes  convenient  to 
write  the  equation  of  the  circle 
solved  for  y  in  the  form 


y=  ±^i 


(2) 


Fig.  52. 


-The  Definition  of  the  Cir- 
cular Functions. 


This  gives,  for  each  value  of  x, 
the  two  corresponding  equal  and 
opposite  ordinates. 
To  translate  the  circle  of  radius  a  so  that  its  center  shall  be  the 
point  {h,  k),  it  is  merely  necessary  to  write 


(x-h)2+  (y  -k)2  =  a^ 


(3) 


This  is  the  general  equation  of  any  circle  in  the  plane  xy,  for  it 
locates  the  center  at  any  desired  point  and  provides  for  any 
desired  radius  a. 

Exercises 

1.  Write  the  equations  of  the  circles  with  center  at    the   origin 
having  radii  3,  4,  11,  V^  respectively. 

94 


§43]   THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS      95 

2.  Write  the  equation  of  each  circle  described  in  exercise  1  in 
the  form  y  =  ±  \/a^'—  x^. 

3.  Which  of  the  following  points  lie  on  the  circle  x^  -\-  y^  ==  169 
(5,  12),  (0,  13),  (  -  12,  5),  (10,  8),  (9,  9),  (9,  10)? 

4.  Which  of  the  following  points  lie  inside  and  which  lie  outside 
of  the  circle  x^  +  y''  =  100:  (7,  7)^  (10,  0),  (7,  8),  (6,  8),  (  -  5,  9), 
(  -  7,   -  8),  (2,  3),  (10,  5),  (V40,  Vso),  (Vl9,  9)? 

42.  The  Equation,  x2  +  y2+  2gx  +  2fy  +  c  =  o  (1) 
may  be  put  in  the  form  (3) .     For  it  may  be  written 

x^  +  2gx  -\-g^  +  y^  +  2fy  ^  p  =  g^  ^  p  -  c, 
or,  

{X    +  gy  H-  (7/  +/)2  =    {^/g^J^p-  cY  (2) 

which  represents  a  circle  of  radius \/^^  -\-p  --c  whose  center  is  at 
the  point  {- g,  -/).  In  case  g^ -^ p  —  c  <  0,  the  radical 
becomes  imaginary,  and  the  locus  is  not  a  real  circle;  that  is, 
coordinates  of  no  points  in  the  plane  xy  satisfy  the  equation.  If 
the  radical  be  zero,  the  locus  is  a  single  point. 

43.  Any  equation  of  the  second  degree,  in  two  variables,  lacking 
the  termxy  and  having  like  coefficients  in  the  terms  x^  andy^,  repre- 
sents a  circle,  real,  null  or  imaginany.  The  general  equation  of 
the  second  degree  in  two  variables  may  be  written: 

ax'  +  67/2  +  2hxy  -^  2gx+ 2fy  +  c  =  0  (3) 

for,  when  only  two  variables  are  present, there  can  be  present  three 
terms  of  the  second  degree,  two  terms  of  the  first  degree,  and  one 
term  of  the  zeroth  degree.  When  a  =  b  and  A  =  0  this  reduces 
to  <1)  above  on  dividing  through  by  a. 

Exercises 

Find  the  centers  and  the  radii  of  the  circles  given  by  the  following 
equations : 

1.  x^  -\-  y^  =25.     Also  determine  which  of  the  following  points 
are  on  this  circle:  (3,  4),  (5,  5),  (4,  3),  (-3,   -4),  (-3,  4),  (5,   0), 

(2,  V21). 

2.  X2    +   7/2    =    H^, 

3.  x2  +  2/2  -  4  =  0. 


96  ELEMENTARY  MATHEMATICAL  ANALYSIS        [§44 

4.  a;2  +  ?/'  -  36  =  0. 
6.  a;2  +  ?/2  +  2a;  =  0. 


6.  2/  =  ±  VIGO  —  x^.     Also  find  the  slope  of  the  diameter  through 
the  point  (5,  12).     Find  the  slope  of  the  tangent  at  (5,  12). 

7.  9  -  a;2  -  2/2  =  0. 

8.  a;2  +  1/2  -  6?/  =  16. 

9.  a;2  -  2a;  +  7/2  -  6?/  =  15. 

10.  {x  +  a)2  +  (2/  -  6)2  =  50. 

11.  x2  +  i/2  +  6x  -  2?/  =  10. 

12.  x2  +  2/2  -  4a;  +  6?/  =  12. 

13.  a;2  +  2/2  -  4a;  -  82/  +  4  =  0. 

14.  3a;2  +  82/^  +  6a;  +  122/  -  60  =  0. 

15.  Is  a;2  +  22/2  +  3a;  —  42/  —  12  =  0  the  equation  of  a  circle  ? 
Why? 

16.  Is  2a;2  +  22/2  -  3a;  +  42/  -  8  =  0  the  equation  of  a  circle? 
Why? 

44.  Angular  Magnitude.     By  the  magnitude  of  an  angle  is 

meant  the  amount  of  rotation  of  a  line  about  a  fixed  point.  If 
a  line  OA  rotate  in  the  plane  XY  about  the  fixed  point  0  to  the 
position  OP,  the  line  OA  is  called  the  initial  side  and  the  line  OF  is 
called  the  terminal  side  of  the  angle  AOF.  The  notion  of  angular 
magnitude  as  introduced  in  this  definition  is  more  general  than 
that  used  in  elementary  geometry.  There  are  two  new  and  very 
important  consequences  that  follow  therefrom: 

(1)  Angular  magnitude  is  unlimited  in  respect  to  size — that  is, 
it  may  be  of  any  amount  whatsoever.  An  angular  magnitude  of 
100  right  angles,  or  twenty-five  complete  rotations  is  quite  as 
possible,  under  the  present  definition,  as  an  angle  of  smaller 
amoujit. 

(2)  Angular  magnitude  exists,  under  the  definition,  in  two 
opposite  senses — for  rotation  may  be  clockwise  or  anti-clockwise. 
As  is  usual  in  mathematics,  the  two  opposite  senses  are  distin- 
guished by  the  terms  positive  and  negative.  In  Fig.  53,  AOP\, 
AOP2,  AOP3,  AOPi  are  positive  angles.  In  designating  an 
angle  its  initial  side  is  always  named  first.  Thus,  in  Fig.  53, 
AOPi  designates  a  positive  angle  of  initial  side  OA.  PiOA 
designates  a  negative  angle  of  initial  side  OPi. 


§451    THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS      97 


Fig.  53. — Triangles  of  Reference 
{ODiPu  OD2P2,  etc.)  for  Angles  d 
of  Various  Magnitude. 


In  Cartesian  coordinates,  OX  is  usually  taken  as  the  initial 
line  for  the  generation  of  angles.  If  the  terminal  side  of  any  angle 
falls  within  the  second  quadrant,  it  is  said  to  be  an  "angle  of  the 
second  quadrant,"  etc. 

Two  angles  which  differ  by 
any  multiple  of  360°  are  called 
congruent  angles.  We  shall 
find  that  in  certain  cases  con- 
gruent angles  may  be  substi- 
tuted for  each  other  without 
modifying  results. 

The  theorem  in  elementary 
geometry,  that  angles  at  the 
center  of  a  circle  are  propor- 
tional to  the  intercepted  arcs, 
holds  obviously  for  the  more 
general  notion  of  angular  mag- 
nitude here  introduced. 

45.  Units  of  Measure.  Angular  magnitude,  like  all  other 
magnitudes,  must  be  measured  by  the  appHcation  of  a  suitable 
unit  of  measure.     Four  systems  are  in  common  use: 

(1)  Right  Angle  System.  Here  the  unit  of  measure  is  the  right 
angle,  and  all  angles  are  given  by  the  number  of  right  angles  and 
fraction  of  a  right  angle  therein  contained.  This  unit  is  f amihar 
to  the  student  from  elementary  geometry.  A  practical  illus- 
tration is  the  scale  of  a  mariner's  compass,  in  which  the  right  angles 
are  divided  into  halves,  quarters  and  eighths. 

(2)  The  Degree  System.  Here  the  unit  is  the  angle  corre- 
sponding to  g^lxr  of  a  complete  rotation.  This  system,  with  the 
sexagesimal  sub-divisions  (division  by  60ths)  into  minutes 
and  seconds,  is  familiar  to  the  student.  This  system  dates  back 
to  remote  antiquity.  It  was  used  by,  if  it  did  not  originate  among, 
the  Babylonians. 

(3)  The  Hour  System.  In  astronomy,  the  angular  magnitude 
about  a  point  is  divided  into  24  hours,  and  these  into  minutes 
and  seconds.  This  system  is  familiar  to  the  student  from  its 
analogous  use  in  measuring  time. 

(4)  The  Radian  or  Circular  System.     Here  the  unit  of  measure 
7 


98 


ELEMENTARY  MATHEMATICAL  ANALYSIS 


§45 


is  an  angle  such  that  the  length  of  the  arc  of  a  circle  described  about 
the  vertex  as  center  is  equal  to  the  length  of  the  radius  of  the 
circle.  This  system  of  angular  measure  is  fundamental  in  me- 
chanics, mathematical  physics  and  pure  mathematics.  It  must 
be  thoroughly  mastered  by  the  student.  The  unit  of  measure  in 
this  system  is  called  the  radian.     Its  size  is  shown  in  Fig.  54. 


O  Radius 

Fig.  54. — Definition    of    the    Radian.     The    Angle    AOP  is  one  Radian. 


Inasmuch  as  the  radius  is  contained  27r  times  in  a  circumference, 
we  have  the^  equivalents: 

27r  radians  =  360°. 
or,  1  radian  =  57°  17'  44".8  =  57°  17'.7  =  57°.3. 

1  degree  =  0.01745  radians. 

The  following  equivalents  are  of  special  importance: 

a  straight  angle  =  tt  radians. 

IT 


a  right  angle  =  ^  radians. 


60°  =  „  radians. 


45^ 


radians. 


30°  =  a  radians, 
o 


§46]    THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS      99 

There  is  no  generally  adopted  scheme  for  writing  angular  magni- 
tude in  radian  measure.  We  shall  use  the  superior  Roman  letter 
"^"  to  indicate  the  measure,  as  for  example,  18°  =  O-SHIG"". 

Since  the  circumference  of  a  circle  is  incommensurable  with  its 
diameter,  it  follows  that  the  number  of  radians  in  an  angle  is 
always  incommensurable  with  the  number  of  degrees  in  the  angle. 

The  speed  of  rotating  parts,  or  angular  velocities,  are  usually 
given  either  in  revolutions  per  minute  (abbreviated  ''r.p.m.") 
or  in  radians  per  second. 

46.  Uniform  Circular  Motion.  Suppose  the  line  OP,  Fig.  52, 
is  revolving  counter-clockwise  ¥  per  second,  the  angle  AOP 
in  radians  is  then  kt,  t  being  the  time  required  for  OP  to  turn  from 
the  initial  position  OA.  If  we  call  d  the  angle  AOP,  we  have  B  =  kt 
as  the  equation  defining  the  motion.  The  following  terms  are 
in  common  use: 

1.  The  angular  velocity  of  the  uniform  circular  motion  is  k 
(radians  per  second). 

2.  The  amplitude  of  the  uniform  circular  motion  is  a . 

3.  The  period  of  the  uniform  circular  motion  is  the  number  of 
seconds  required  for  one  revolution. 

4.  The  frequency  of  the  uniform  circular  motion  is  the  number 
of  revolutions  per  second. 

Sometimes  the  unit  of  time  is  taken  as  one  minute.  Also  the 
motion  is  sometimes  clockwise  or  negative. 


Exercises 

1.  Express  each  of  the  following  in  radians:  135°,  330",  225°,  15°, 
150°,  75°,  120°.     (Do  not  work  out  in  decimals;  use  ir). 

2.  Express  each  of  the  following  in  degrees  and  minutes:    0.2*", 

7rV5,  |-x^  V. 

3.  How  many  revolutions  per  minute  is  20  radians  per  second? 

4.  The  angular  velocity,  in  radians  per  second,  of  a  36-inch 
automobile  tire  is  required,  when  the  car  is  making  20  miles  per  hour. 

6.  What  is  the  angular  velocity  in  radians  per  second  of  a  6-foot 
drive-wheel,  when  the  speed  of  the  locomotive  is  50  miles  per  hour? 

6.  The  frequency  of  a  cream  separator  is  6800  r.p.m.  What  is 
its  period,  and  velocity  in  radians? 


100        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§47 

7.  A  wheel  is  revolving  uniformly  SO'^  per  second.  What  is  its 
period,  and  frequency? 

8.  The  speed  of  the  turbine  wheel  of  a  5-h.p.  DeLaval  steam  turbine 
is  30,000  r.p.m.  What  is  the  angular  velocity  in  radians  per 
second? 

47.  The  Circular  or  Trigonometric  Functions.  To  each  point 
on  the  circle  x^  -\-y^  =  a^  there  corresponds  not  only  an  abscissa  and 
an  ordinate,  but  also  an  angle  6  <  360°,  as  shown  in  Figs.  52  and  53. 
This  angle  is  called  the  direction  angle  or  vectorial  angle  of  the 
point  P.  When  6  is  given^  x,  y  and  a  are  not  determined,  but  the 
ratios  y /a,  xja,  y /x,  and  their  reciprocals,  a/y,  a/x,  x/y  are  de- 
termined. Hence  these  ratios  are,  by  definition,  fimctions  of  6. 
They  are  known  as  the  circular  or  trigonometric  functions  of  d, 
and  are  named  and  written  as  follows: 


Fimction  of  d. 

Name. 

Written 

y/a. 

sine  of  6. 

sin  e. 

x/a. 

cosine  of  d. 

cos  e. 

y/x. 

tangent  of  d. 

tan  d. 

x/y. 

cotangent  of  6. 

cot  0. 

a/x. 

secant  of  d. 

sec  d. 

a/y. 

cosecant  of  d. 

esc  0, 

The  circular  functions  are  usually  thought  of  in  the  above  order: 
that  is,  in  such  order  that  the  first  and  last,  the  middle  two,  and 
those  intermediate  to  these,  are  reciprocals  of  each  other. 

The  names  of  the  six  ratios  must  be  carefully  committed  to 
memory,  They  should  be  committed,  using  the  names  of  x,  y, 
and  a  as  follows: 

Ratio.  Written, 

ordinate  /radius.  sin  0,  ] 

abscissa /radius.  cos  0, 

ordinate /abscissa.  tan  0. 

abscissa /ordinate.  cot  0. 

radius /abscissa.  sec  0. 

radius /ordinate.  esc  0, 

The  right  triangle  POD  of  sides  x,  y  and  a,  whose  ratios  give  the 
functions  of  the  angle  XOP,  is  often  called  the  triangle  of  reference 


THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS,   101 

for  this  angle.  It  is  obvious  that  the  size  of  the  triangle  of  refer- 
ence has  no  effect  of  itself  upon  the  value  of  the  functions  of  the 
angle.  Thus  in  Fig.  53  either  PiODi  or  Pi'ODi  may  be  taken  as 
the  triangle  of  reference  for  the  angle  ^i.  Since  the  triangles  are 
similar  we  have. 


ODi        ODi'  OPi        OPi' 

etc.,  which  shows  that  identical  ratios  or  trigonometric  functions  of 
d  are  derived  from  the  two  triangles  of  reference. 

48.  Elaborate  means  of  computing  the  six  functions  have  been 
devised  and  the  values  of  the  functions  have  been  placed  in 
convenient  tables  for  use.  The  functions  are  usually  printed 
to  3,  4,  5  or  6  decimal  places,  but  tables  of  8,  10  and  even  14  places 
exist.  The  functions  of  only  a  few  angles  can  be  computed  by 
elementary  means;  these  angles,  however,  are  especially  important. 

(1)  The  Functions  of  30°.  In  Fig.  55a,  if  angle  AOB  be  30°, 
angle  ABO  must  be  60°.  Therefore,  constructing  the  equilateral 
triangle  BOB',  each  angle  of  triangle  BOB'  is  60°,  and 

y  =  AB  =  ^-BB'  =  i-a 
Therefore, 

sin  30°  =  ^  =  ^  =  1/2 
a         a 

Also : 

OA  =  V~aB2  _  j^2  =  Va2  -  |a2  =  ^a  Vs 

Therefore, 

sin  30°  =  1/2. 

_  i  a  \/3  _  \/3 


cos  30' 
tan  30°  = 


a  2 

iaVS         3 
^^*3«°  =  tan^°=^^ 


sec  30°  =     ._  „^o 


1       ^  2\/3 
cos  30°  3" 


CSC  30°  =    .  -ono  =  2 
sm  30 


102        ET.EMENTARY  MATHEMATICAL  ANALYSIS 


(48 


(2)  Functions  of  45°.  In  the  diagram,  Fig.  556,  the  triangle 
OAB  is  isosceles,  so  that  y  =  x,  and  a-  =  x-  +  y^  =  2x^.  It 
follows  that  a  =  x-\/2  =  y-\/2. 


-j£ 


Fig.   55.— Triangles  of  Reference  for  Angles  of  30°,  45°  and  60°. 

Therefore: 


sin  45°  = 

y 

V2 

y\/2 

X 

2 
V2 

cos  45    = 

x-\/2  ■ 

2 

tan  45°  = 

1=1 

cot  45°  = 

1 
tan  45° 

=  1 

sec  45°-= 

1 

cos  45° 

=  V2 

CSC  45°  = 

1 

'          A  SO 

-  V2 

(3)  Functions  of  60°.  In  the  diagram.  Fig.  55c,  construct  the 
equiangular  triangle  OBB';  then  it  is  seen  that,  as  in  case  (1) 
above, 

OA  =  h'OR'  =  |-a 
and 

y  =  Va^  -i-ct^  =  h'a\/S 

Therefore:  .    ^^o      2*a'\/3         \/3 

sm  oU  = =      ^ 

a  I 


§49]    THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS    103 

cos60°  =  i^=l/2 
a 

tan  00°=  t^^^   =  ^3 


cot  60°  = 
sec  60°  = 


1        ^  V3_ 
tan  60°  3 

1 


cos  60° 

1  2x/3 


•    CSC  60    =  ".    ^^o 

sin  60  3 

49.  Graphical  Computation  of  Circular  Functions.  Approximate 
determination  of  the  numerical  values  of  the  circular  functions  of 
any  given  angle  may  be  made  graphically  on  ordinary  coordinate 
paper.  .  Locate  the  vertex  of  the  angle  at  the  intersection  of  any 
two  lines  of  the  squared  paper,  form  Ml.  Let  the  initial  side  of 
the  angle  coincide  with  one  of  the  rulings  of  the  squared  paper 
and  lay  off  the  terminal  side  of  the  angle  by  means  of  a  protractor. 
If  the  sine  or  cosine  is  desired,  describe  a  circle  about  the  vertex 
of  the  angle  as  center  using  a  radius  appropriate  to  the  scale  of 
the  squared  paper — for  example,  a  radius  of  5  cm.  on  coordi- 
nate paper  ruled  in  centimeters  and  fifths  (form  Ml)  permits 
direct  reading  to  1/25  of  the  radius  a  and,  by  interpolation,  to 
1  /lOO  of  the  radius  a.  The  abscissa  and  ordinate  of  the  point 
of  intersection  of  the  terminal  side  of  the  angle  and  the  circle  may 
then  be  read  and  the  numerical  value  of  sine  and  cosine  computed 
by  dividing  by  the  length  of  the  radius. 

If  the  numerical  value  of  the  tangent  or  cotangent  be  required, 
the  construction  of  a  circle  is  not  necessary.  The  angle  should 
be  laid  off  as  above  described,  and  a  triangle  of  reference  con- 
structed. To  avoid  long  division,  the  abscissa  of  the  triangle  of 
reference  may  be  taken  equal  to  50  or  100  mm.  for  the  determina- 
tion of  the  tangent  and  the  ordinate  may  be  taken  equal  to  50 
or  100  mm.  for  the  determination  of  the  cotangent. 

The  following  table  (Table  III)  contains  the  trigonometric 
functions  of  acute  angles  for  each  10°  of  the  argument. 


104        ELEMENTARY  MATHEMATICAL  ANALYSIS        (§19 


Tablk  III 
Natural  Trigonometric  Functions  to  Two  Decimal  Places 


0 

d' 

sin  e 

cos  d 

tan  d 

cot  d 

sec  d 

CSC  d 

0.00 

0.00 

1.00 

0.00 

00 

1.00 

00 

10 

0.17 

0.17 

0.98 

0.18 

5.67 

1.02 

5.76 

20 

0.35 

0.34 

0.94 

0.36 

2.75 

1.06 

2.92 

30 

0.52 

0.50 

0.87 

0.58 

1.73 

1.15 

2.00 

40 

0.70 

0.64 

0.77 

0.84 

1.19 

1.31 

1.56 

50 

0.87 

0.77 

0.64 

1.19 

0.84 

1.56 

1.31 

60 

1.05 

0.87 

0.50 

1.73 

0.58 

2.00 

1.15 

70 

1.22 

0.94 

0.34 

2.75 

0.36 

2.92 

1.06 

80 

1.40 

0.98 

0.17 

5.67 

0.18 

5.76 

1.02 

90 

1.57 

1.00 

0.00 

00 

0.00 

00 

1.00 

The  most  important  of  these  results  are  placed  in  the  following 
table: 


0° 

30° 

45° 

60° 

90° 

Sine 

Cosine 

0 

1 

0 

1/2 

a/3 
2 

V3 
3 

V2 
2 

V2 
2 

1 

V3 
2 

1/2 
V3 

1 
0 

00 

Tangent 

V2   = 

1.4142 

! 

Vs  - 

=  1.7321 

Exercises 

1.  Find  by  graphical  construction  all  the  functions  of  15°. 
Note. — A  protractor  is  not  needed  as  angles  of  45°  and  30°  may  be 

constructed. 

2.  Find  tan  60°.     Compare  with  the  value  found  above  in  §48. 

3.  Lay  off  angles  of  10°,  20°,  30°,  40°,  with  a  protractor  and  deter- 
mine graphically  the  sine  of  each  angle,  and  record  the  results  in  a 
suitable  table. 

4.  Find  the  sine,  cosine,  and  tangent  of  75°. 

5.  Which  is  greater,  sec  40°  or  esc  50°? 

6.  Determine  the  angle  whose  tangent  is  1/2. 

7.  Find  the  angle  whose  sine  is  0.6. 


§50]    THE  CIRCLE  AND  THE  CIRCUT.AR  FTTNCTIONS    105 

8.  Which  is  greater,  sin  40°  or  2sin  20°? 

9.  Does  an  angle  exist  whose  tangent  is  1,000,000?  What  is  its 
approximate  value? 

50.  Signs  of  the  Functions.  The  circular  functions  have,  of 
course,  the  algebraic  signs  of  the  ratios  that  define  them.  Of 
the  three  numbers  entering  these  ratios,  the  distance  or  radius 
a  may  always  be  taken  as  positive.  It  enters  the  ratios,  there- 
fore as  an  always  signless,  or  positive  number.  The  abscissa 
and  the  ordinate,  x  and  y,  have  the  algebraic  signs  appropriate 
to  the  quadrants  in  which  P  falls.  The  student  should  deter- 
mine the  signs  of  the  functions  in  each  quadrant,  as  follows: 
(See  Fig.  53.) 


First 
quadrant 

Second 
quadrant 

Third 
quadrant 

Fourth 
quadrant 

Sine 

+ 
+ 
+ 

+ 

+ 

+ 

Cosine 

Tangent 

Of  course  the  reciprocals  have  the  same  signs  as  the  original 
functions. 

The  signs  are  readily  remembered  by  the  following  scheme: 


Sine 

Cos 

ime 

Tar 

igent 

+     + 

- 

+    ' 

- 

+ 

— 

- 

- 

+ 

+ 

- 

Cose 

cant 

Sec 

ant 

Cota 

Qgent 

The  following  scheme  is  of  value  in  remembering  the  circular 
functions  and  their  signs  in  the  different  quadrants :  Place  on  the 
same  line  the  variables  and  functions  of  the  same  algebraic  signs, 
thus  : 

Ordinate  .  .  y  .  .  sin  ^  .  .  esc  ^ 
Abscissa  .  .  x  .  .  cos  d  .  .  sec  0 
Slope    .    .    .  m  .    .  tan  d  .   .  cot  6 


lOG        ELEMENTARY  MATHEMATICAL  ANALYSLS         fSf)! 


The  above  scheme  associates  the  signs  of  the  functions  with  the 
coordinates  (x,  y)  of  the  point  P  and  the  slope  of  the  hne  OP 
for  each  of  its  four  positions  in  Fig.  53. 

51.  Triangles  of  reference,  geometrically  similar  to  those  in 
Fig.  55  for  angles  of  30°,  45°,  and  60°  exist  in  each  of  the  four 
quadrants,  namely,  when  the  hypotenuse  and  a  leg  of  the  triangle 
of  reference  in  these  quadrants  are  both  either  parallel  or  perpen- 
dicular to  a  hypotenuse  and  leg  of  the  triangle  in  the  first  quad- 
rant— then  an  acute  angle  of  one  must  equal  an  acute  angle  of 
the  other  and  the  triangles  must  be  similar.  The  numerical 
values  of  the  functions  in  the  two  quadrants  are  therefore  the 
same.  The  algebraic  signs  are  determined  by  properly  taking 
account  of  the  signs  of  the  abscissa  and  the  ordinate  in  that 
quadrant.     Thus  the  triangle  of  reference  for  120°  is  geometri- 


cally similar    to    that  for    60°.     Hence,    sin    120°  = 
cos  120°  =  -  1/2  and  tan  120°  =  -a/S. 


V3 


but 


Exercises 

1.  The  student  is  to  fill  in  the  blanks  in  the  following  table  with 
the  correct  numerical  value  and  the  correct  sign  of  -each  function : 


Function      120°     135°    150° 

1            1 

210°  1  225° 

240° 

300°    315°    330° 

Sin 

i           i 
'         ■  i 

Cos           1 

i        ! 
1        1 

1 

i 

Tan 

Cot           1 

! 

!        i 

Sec           i 

. 

i 
t 

1 

Csc 

i        1 

j 

2.  Write  down  the  functions  of  390°  and  405°. 

3.  The  tangent  of  an  angle  is  1.     What  angle  <  360°  may  it  be? 

4.  Cos  ^  =  -  1/2.     What  two  angles  <  360°  satisfy  the  equation? 
6.  Sec  6=2.     Solve  for  all  angles  <  360°. 

6.  Csc  e  =  -  \/2.     Solve  for  6  <  360°. 


§52]    THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS    107 


52.  Functions  of  0°  and  90°.  In  Fig.  52  let  the  angle  AOP 
decrease  toward  zero,  the  point  P  remaining  on  the  circumference 
of  radius  a.  Then  y  or  PD  decreases  toward  zero.  Therefore, 
sin  0°  =  0.  Also,  x  or  OD  increases  to  the  value  a,  so  that  the 
ratio  xja  becomes  unity,  or  cos  0°  =  1.  Likewise  the  ratio 
y  fx  becomes  zero,  or  tan  0°  =  0. 

The  reciprocals  of  these  functions  change  as  follows :  As  the  angle 
AOP  becomes  zero,  the  ratio  afy  increases  in  value  without  limit, 
or  the  cosecant  becomes  infinite.  In  symbols  (see  §23) 
CSC  0°  =  c».     Likewise,  cot  0°  =  oo,  but  sec  0°  =  1. 

In  a  similar  way  the  functions  of  90°  may  be  investigated.  The 
results  are  given  in  the  following  table: 


Angle 

From                From 

From 

From 

0°  to  90°    j  90°  to  180° 

1 

180°  to  270° 

270°  to  360° 

Sin 

1 
Oto  +  1     :+    1  to      0 

Cos 

+  Uo    0           0  to-  1 

Tan 

Oto+oo;  -  ootoH-  1 

Cot 

+  oo  to     0 

0  to  -  00 

Soc 

-f-   1  to+  00 

—  CO  to  —  1 

Cso 

+  ooto  +  1      +  1    to+oo 

i                           1 

The  student  is  to  supply  the  results  for  the  last  two  columns. 

53.  Fimdamental  Relations.  The  trigonometric  functions  are 
not  independent  of  each  other.  Because  of  the  relation  x^  +  y^ 
—  a^  it  is  possible  to  compute  the  numerical  or  absolute  value  of 
five  of  the  functions  when  the  value  of  one  of  them  is  given.  This 
may  be  accomplished  by  means  of  the  fundamental  formulas  de- 
rived below: 

Divide  the  members  of  the  equation: 


by  a\    Then 


or, 


a;2  _^  2/2  =  ^2 

sin2  e  +  cos2  e  =  1 


(1) 


(2) 


108        ELEMENTARY  MATHEMATICAL  ANALYSIS 
Likewise  divide  (1)  through  by  a;^:  then 

or, 

sec2  ^  =  1  +  tan2  6 

Also  divide  (1)  through  by  i/^:  then 


§53 


or, 

Also,  since 


csc2  (9=1  +  cot2  e 


(3) 


(4) 


we  obtain^: 
and  likewise 


X  X 

a 


tan^  = 


cot  ^  = 


cose 
cos  d 


sm  e 

sin  =1 /esc 


=l/8ec 


tan  =l/cot 


cot 


+  tan  2 


(5) 
(6) 


tan 


csc-=l  +  cot- 


FiG.  56. — Diagram  of  the  Relations  between  the  Six  Circular  Functions. 

Formulas  (2)  to  (6)  are  the  fundamental  relations  between  the  six 
trigonometric  functions.  The  formulas  must  be  committed  to 
memory  by  the  student. 

The  above  relations  between  the  expressions  may  be  illustrated 


§53]    THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS    109 

by  a  diagram  as  in  Fig.  56.     The  simpler  or  reciprocal  relations  are 
shown  by  the  connecting  lines  drawn  above  the  functions. 

The  reciprocal  equations  and  the  formulas  (2),  (3)  and  (4)  are 
sufficient  to  express  the  absolute  or  numerical  value  of  any  function 
of  any  angle  in  terms  of  any  other  function  of  that  angle.  The 
algebraic  sign  to  be  given  the  result  must  be  properly  selected  in 
each  case  according  to  the  quadrant  in  which  the  angle  lies. 


Exercises 

All  angles  in  the  following  exercises  are  supposed  to  be  less  than 
ninety  degrees. 

1.  Sin  d  =  1/5.     Find  cos  d  and  tan  d. 

Draw  a  right  triangle  whose  hypotenuse  is  5  and  whose  altitude  is 
1  so  that  the  base  coincides  with  OX.  In  other  words,  make  a  =  5 
and  y  =  1  in  Fig.  57.  Calculate  x  =  V25  —  1  =  2V^  and  write 
down  all  of  the  functions  from  their  definitions. 


Fig.  57. 


O  X  A 

-Triangle  of  Reference  for  d  and  Complement  of  6. 


2.  Cos  0  =  1/3.     Find  esc  d. 

Take  a  =  3  and  a;  =  1  in  Fig.  57.     Find  y  and  then  write  down  the 
functions  from  their  definitions. 

3.  Tan  6=2.     Find  sin  d. 

Take  a;  =  1  and  y  =  2  in  Fig.  57,  and  calculate  a  and  then  write 
down  the  functions  from  their  definitions. 

4.  Sec  0  =  10.     Find  esc  d. 

Take  a  =  10  and  a;  =  1  and  compute  y. 

6.  Find  the  values  of  all  functions  of  d  if  cot  0  =  1.5. 

6.  Find  the  functions  of  0  if  cos  0  =  0.1. 

7.  Find  the  values  of  each  of  the  remaining  circular  functions  in 
each  of  the  following  cases: 


110        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§53 


Va^  +  6- 


(a)  sin  6  =  5/13.  (c^)  tan  6  =  3/4.  (g)  tan  d  =  m. 

(6)   cos  d  =  4/5.  (e)   sec  0  =  2.  (/i)    sin 

(c)   sec  0  =  1.25.  (/)    tan  6  =  1/3. 

Show  that  the  following  equahties  are  correct: 

8.  Tan  0-cos  6  =  sin  0. 

9.  Sin  0cot  &sec  0  =  1. 

10.  (Sin  0  +  cos  0)2  =  2-sin  0-cos  0  +  1. 

11.  Tan  0  +  cot  0  =  sec  0csc  6. 

12.  Express  each  trigonometric  function  in  terms  of  each  of  the 
others;  i.e.,  fill  in  all  blank  spaces  in  the  following  table: 


sin 

cos 

tan 

cot 

sec 

CSC 

sin 

sin 

1 

CSC 

cos 

cos 

t 

J 

sec 

tan 

! 

tan 

1 
cot 

cot 

1 
tan 

cot 

sec 

1 

COS 

sec 

CSC 

1 

sin 

CSC 

The  following  exercises  refer  to  angles  <  360°  of  any  quadrant : 

13.  If  sin  0  =  —3/4  and  tan  0  is  positive,  find  the  remaining  five 
functions. 

14.  If  cos  0  =  12/13  and  sin  0  is  negative,  find  the  remaining 
five  functions. 

16.  If  tan  0  =  —  V 3  and  cos  0  is  negative,  find  the  remaining  func- 
tions of  0. 

16.  If  cos  0  =  —  1/3  and  sin  0  is  positive,  find  the  remaining 
functions. 


§54]    THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS    111 


(-k.h)  Pi 


Pi(k.h) 


P(h.ky 


17.  If  tan  0  =  5/12  and  sec  d    is   negative,    find  the  remaining 
functions  of  6. 

18.  If  sin  6  =  3/5  and  tan  d  is  negative,  find  the  remaining  func- 
tions of  6. 

54.  Functions  of  Comple- 
mentary Angles.  Complementary 
angles  are  defined  as  two  angles 
whose  sum  is  90° .  Supplementary 
angles  are  two  angles  whose  sum 
is  180°. 

Let  d  be  an  angle  of  the  first 
quadrant,  and  draw  the  angle 
(90°-  6)  of  terminal  side  OPi,  as 
shown  in  Fig.  58.  Let  P  and  Pi 
lie  on  a  circle  of  radius  a.  Let 
the  coordinates  of  the  point  P  be 
(/i,  k),  then  Pi  is  the  point  {k,  h). 
Hence  PiDi/OPi  =  h/a  = 
sin  (90°  -  6).  But  from  the  tri- 
angle PDO,  h  /a  =  cos  6.    Hence 


(k,-h) 


Fig. 


A  (k..h) 


58. — Triangles  of  Reference 
for  6 ,  and  6  combined  with  an 
Odd  Number  of  Right  Angles. 


Likewise, 


sin  (90°  —  6)  =  cos 

tan  (90°  -  6)  =  cot 
sec  (90°  -  d)  =  CSC 


These  relations  explain  the  meaning  of  the  words  cosine,  cotangent, 
cosecant,  which  are  merely  abbreviations  for  complement's  sine, 
complement's  tangent,  etc.  Collectively,  cosine,  cotangent,  cosecant 
are  called  the  co-functions.    Likewise  from  Fig.  58: 

cos  (90°  —  d)  =  sin  6 
cot  (90°  -  6)  =  tan  6 
CSC  (90°  —  ^)  =  sec  ^ 

Later  it  will  be  shown  that  the  above  relations  hold  for  all 
values  of  6,  positive,  or  negative. 

55.  Graph  of  the  Sine  and  Cosine.  In  rectangular  coordinates 
we  can  think  of  the  ordinate  y  of  a  point  as  depending  for  its  value 
upon  the  abscissa  or  x  of  that  point  by  means  of  the  equation  y  = 
sin  a;,  provided  we  think  of  each  value  of  the  abscissa  laid  off  on 


112 


ELEMENTARY  MATHEMATICAL  ANALYSIS 


[§55 


the  X-axis  as  standing  for  some  amount  of  angular  magnitude. 
Therefore  the  equation  y  =  sinx  must  possess  a  graph  in  rectangu- 
lar coordinates.  In  order  to  produce  the  graph  of  2/  =  sin  x,  it  is 
best  to  lay  off  the  angular  measure  x  on  the  X-axis  in  such  a  manner 
that  it  may  conveniently  be  thought  of  in  either  radian  or  degree 
measure.  If  we  suppose  that  a  scale  of  inches  and  tenths  is  in  the 
hands  of  the  reader  and  that  a  graph  is  required  upon  an  ordinary 
sheet  of  unruled  paper  of  letter  size  (8|  X  11  inches),  then  it  will 
be  convenient  to  let  1  /5  inch  of  the  horizontal  scale  of  the  X-axis 
correspond  to  10°  or  t0  7r/18  radians   of  angular  measure.     To 


Y 

/I.., 

^ 

c 

^ 

— 1 

_^ 

__ 









P_, 



, 

^ 

Wr 

- 

- 

7" 

^ 

r 

— 

- 

J 

H 

s. 

ST 

— 

— 

— 

— 

- 

- 

L 

- 

- 

- 

- 

- 

- 

- 

4r 

/ 

s 

% 

/ 

S 

4/ 

\ 

Z') 

o 

\ 

l^ 

^    AVJ 

/  1 

\ 

~^ 

-A 

\ 

/ 

^^ 

1 

S 

/ 

y 

J 

s 

/ 

yy 

/ 

— 

— 

— 

— 

— 

— 

- 

— 

— 

— 

— 

- 

— 

— 

- 

- 

s 

s 

- 

- 

- 

-- 

;^ 

y 

- 

_^.^^ 

— 

J^ 

^ 

— 

^ 

= 

^ 

^ 

= 

= 

^ 

_ 

_ 

_ 

_ 

^ 

2 

rr— 

^ 

^ 

^ 

^ 

^ 

s 

^ 

T- 

^ 

= 

^ 





.     Fig.  59. — Construction  of  the  Sinusoid. 

accomplish  this,  the  length  of  one  radian  must  be  1.15  inches  (t.e., 
18/57r  inch),  which  length  must  be  used  for  the  radius  of  the  circle 
on  which  the  arcs  of  the  angles  are  laid  off.  Hence,  to  graph 
y  =  sin  X,  draw  at  the  left  of  a  sheet  of  (unruled)  drawing  paper  a 
circle  of  radius  1.15  inches,  as  the  circle  OFB^  Fig.  59.  Take  0  as 
the  origin  and  prolong  the  radius  BO  for  the  positive  portion  OX  of 
the  X-axis.  Subdivide  this  into  1  /5-inch  intervals,  each  corre- 
sponding to  10°  of  angle;  eighteen  of  these  correspond  to  the 
length  TT,  if  the  radius  BO  (1.15  inches)  be  the  unit  of  measure. 
Next  divide  the  F-axis  proportionately  to  sin  x  in  the  following 
manner:  Divide  the  semicircle  into  eighteen  equal  divisions  as 
shown  in  the  figure,  thus  making  the  length  of  each  small  arc 
exactly  1/5  inch.  The  perpendiculars,  or  ordinates,  dropped 
upon  OX  from  each  point  of  division,  divided  by  the  radius  a, 
are  the  sines  of  the  respective  angles.  Draw  lines  parallel  to 
OX  through  each  point  of  division  of  this  circle.  These  cut  the 
F-axis  at  points  A 1,  A  2,   .    .    .   ,  such  that  OAi,  0^2,   ...  are 


§56]    THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS    113 

proportional  to  sin  OBPi,  sin  OBP2,  sin  OBP^,  .  .  .  or  in  the 
general  case,  proportional  to  sin  x  (for  lack  of  room  only  a  few 
of  the  successive  points  Pi,  P2,  P3,  .  •  • ,  of  division  of  the 
quadrant  OP3P5,  are  actually  lettered  in  Fig.  59).  These  are  the 
successive  ordinates  corresponding  to  the  abscissas  already 
laid  off  on  OL.  The  curve  is  then  constructed  as  follows: 
First  draw  vertical  lines  through  the  points  of  division  of  OX; 
these,  with  the  horizontal  lines  already  drawn,  divide  the 
plane  into  a  large  number  of  rectangles.  Starting  at  0  and 
sketching  the  diagonals  (curved  to  fit  the  alignment  of  the  points) 
of  successive  "cornering"  rectangles,  the  curve  OCNTL  is  approxi- 
mated, which  is  the  graph  oi  y  =  sin  x.  This  curve  is  called  the 
sinusoid  or  sine  curve.  The  curve  is  of  very  great  importance  for 
it  is  found  to  be  the  type  form  of  the  fundamental  waves  of  science, 
such  as  sound  waves,  vibrations  of  wires,  rods,  plates  and  bridge 
members,  tidal  waves  in  the  ocean,  and  ripples  on  a  water  surface. 
The  ordinary  progressive  waves  of  the  sea  are,  however,  not  of 
this  shape.  Using  terms  borrowed  from  the  language  of  waves,  we 
may  call  C  the  crest,  N  the  node,  and  T  the  trough  of  the  sinusoid. 

It  is  obvious  that  as  x  increases  beyond  27r^,  the  curve  is  re- 
peated, and  that  the  pattern  OCNTL  is  repeated  again  and  again 
both  to  the  left  and  the  right  of  the  diagram  as  drawn.  Thus  it  is 
seen  that  the  sine  is  a  periodic  function  of  period  27r^,  or  360°. 

The  small  rectangles  lying  along  the  X-axis  are  nearly  squares. 
They  would  be  exactly  equilateral  if  the  straight  line  OA 1  was  equal 
to  the  arc  OPi.  This  equality  is  approached  as  near  as  we  please 
as  the  number  of  corresponding  divisions  of  the  circle  and  of  OX  is 
indefinitely  increased.  In  this  way  we  arrive  at  the  notion  of  the 
slope  of  a  curve  in  mathematics.  In  this  case  we  say  that  the 
slope  of  the  sinusoid  at  0  is  -h  1  and  at  AT  is  —  1,  and  at  L  is  -h  1. 
We  say  that  the  curve  cuts  the  axis  at  an  angle  of  45°  at  0  and 
at  an  angle  of  315°  (or,  —  45°  if  we  prefer)  at  A".  The  slope  at  C 
and  at  T  is  zero. 

The  curve  y  =  a  sin  x  is  made  from  y  =  sin  x  by  multiplying 
all  of  the  ordinates  of  the  latter  by  a.  The  number  a  is  called 
the  amplitude  of  the  s'nusoid. 

56.  Cosine  Curve.  If  0'  be  taken  as  the  origin,  the  curve  CNTL 
is  the  graph  of  y  =  cos  x.    Let  the  student  demonstrate  this  by 

8 


114        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§57 


showing  that  the  distances  BDi,  BD2,  .  .  .  ,  BD4,  ...  in  the 
semicircle  at  the  left  of  Fig.  59  go  through  in  reverse  order  the 
same  sequence  of  values  as  PiDi,  P2D2,  .  ■  ■  ,  and  that  if  the 
origin  be  taken  at  0',  the  successive  ordinates  of  the  sinusoid  to  the 
right  of  O'C  are  equal  to  BDi,  BD2,  .  .  .  respectively,  and  hence 
are  proportional  to  cos  x. 

It  is  best  to  carry  out  the  construction  of  the  sinusoid  upon 
unruled  drawing  paper  as  described  above.  The  curve  can  readily 
be  drawn,  however,  upon  form  M2,  which  is  already  ruled  in 
1  /5-inch  intervals,  or  upon  form  Ml  if  the  radius  of  the  circle  be 
taken  as  2.3  cm.  and  if  2/5  cm.  be  used  on  OX  to  represent 
an  angle  of  10°.  A  much  neater  result  is  obtained  when 
unruled  paper  is  used  for  the  drawing. 

57.  Complementary  Angles.  The  graph  7/2  =  sin  {  —  x)  is 
made  from  7/1  =  sin  x  by  substituting  {—  x)  for  x  in  the  function 


sin  (-x) 


Fig.  go. — Shows  the  Relation  Between  y  =  sin  x  and  y  =  sin  (—x)  and 
Between  y  =  sin  (90°  —  x)  and  y  =  cos  x,  etc. 

sin  x;  that  is,  by  changing  the  signs  or  reversing  the  direction  of 
all  of  the  abscissas  of  the  sinusoid  y  =  sin  x;  or,  in  other  words, 
7/2  =  sin(—  x)  is  the  reflection  of  7/1  =  sin  a:  in  the  F-axis. 
This  is  merely  a  special  case  of  the  general  Theorem  I  on  Loci, 
§24.  The  former  curve  has  a  crest  where  the  latter  has 
a  trough  and  vice  versa,  as  is  shown  by  the  dotted  and  full 
curves  in  Fig.  60.  Now,  if  the  curve  7/2  =  sin  ( —  x)  (the  dotted 
curve  in  Fig.  60)  be  translated  to  the  right  the  distance  7r/2, 
the  resulting  locus  is  the  cosine  curve  y  =  cos  x.  To  translate 
7/2  =  sin  (—  x)  to  the  right  the  distance  7r/2,'  the  constant  7r/2 
must  be  subtracted  from  the  variable  x  in  the  equation  of  the 
curve,  as  already  learned  in  the  last  chapter.  Performing  this 
operation  we  have,  for  the  translated  curve, 


V: 


.in(-[..-y) 


§58]    THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS    115 

(Note  that  7r/2  is  subtracted  from  x  and  not  from  —  x.)     Or, 
removing  the  brackets, 


7/2  =  sm 


(i-) 


But,  as  stated  above,  the  curve  in  its  new  position  is  the  same  as 
the  cosine  curve 

y  =  cosx 
Hence,  for  all  values  of  x: 

sin  (^  —  X )  =  cosx  (1) 

In  the  same  manner  it  can  be  proved  that  cos  (^  —  a:  J  =  sin  x, 

and  the   other  results  of    §54  follow  for  all  values  of  x. 

58.  Trigonometric  Functions  of  Negative  Arguments.  First 
compare  the  curves  ?/i  =  sin  x  and  ?/2  =  sin  (—  x)  as  has  been 
done  in  the  preceding  section,  and  as  is  illustrated  by  Fig.  60. 
The  curve  2/2  =  sin  (—  x)  was  described  as  the  reflection  of  the 
sinusoid  ?/i  =  sin  a;  in  the  F-axis.  It  is  obvious  from  the  figure, 
however,  that  the  dotted  curve  may  also  be  regarded  as  the 
reflection  of  the  original  curve  in  the  X-axis;  for  the  one  has  a 
crest  where  the  other  has  a  trough  and  the  ordinates  of  the  two 
curves  are  everywhere  of  exactly  equal  length  but  opposite  in 
direction.     This  means  that  2/2  =  —  Vi,  or, 

sin  (—  x)  =  —  sin  x  (1) 

for  all  values  of  x. 

If  the  origin  be  taken  at  the  point  0',  Fig.  60,  the  full  curve 
is  the  graph  of  y  =  cos  x.  In  this  case  the  crest  of  the  curve  lies 
al)ove  the  origin  and  the  curve  is  symmetrical  with  respect  to  the 
F-axis.  This  means  that  changing  x  to(— x)in  the  equation 
y  =  cos  x  does  not  modify  the  locus.  Hence  we  conclude  that 
cos  ( —  x)  =  cos  X  (2) 

for  all  values  of  x.     Hence  by  division 

tan  (-x)  =  -tan  X  (3) 

59.  Odd  and  Even  Functions.  A  function  that  changes  sign 
but  retains  the  same  numerical  value  when  the  sign  of  the  variable 
is  changed  is  called  an  odd  function.  Thus  sin  x  is  an  odd  function 
of  X*,  since  sin  (—  x)  =  —  sin  x.     Likewise  x^  is  an  odd  function 


116        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§59 

of  X,  as  are  all  odd  powers  of  x.  Geometrically,  the  graph  of  an 
odd  function  of  x  is  symmetrical  with  respect  to  the  origin  0; 
that  is,  if  P  is  a  point  on  the  curve,  then  if  the  line  OP  be  pro- 
duced backward  through  0  a  distance  equal  to  OP  to  a  point 
P',  then  P'  lies  also  on  the  curve.  The  branches  oi  y  =  x^  in 
the  first  and  third  quadrants  are  good  illustrations  of  this 
property. 

A  function  of  x  that  remains  unaltered  (both  in  sign  and 
numerical  value)  when  the  variable  is  changed  in  sign,  is  called 
an   even  function  of  x.    Examples  are  cos  x,  x^,  x^  —  3x^,  .    .    . 

Most  functions  are  neither  odd  nor  even,  but  mixed,  like 
x^  -\-  sin  X,  x^  -\-  x^,  X  -{■  cos  x,  .    .    . 

Exercises 

1.  Show  from  (1)  and  (2)    §58  and  the  relations  esc  x  =    -. — -» 

sin  X 

=  tan  X,  etc.,  that 

cos  X  '        ' 

(a)  CSC  {—  x)  =  —  esc  X 

(6)  sec  (—  x)  =       sec  x 

(c)  tan  {—  x)  =  —  tan  x 

(d)  cot  (—  x)  =  —  cot  X. 

2.  Is  sin 2  X  an  odd  or  an  even  function  of  a;?  Is  tan^  x  an  odd  or  an 
even  function  of  re? 

3.  Is  the  function  sin  x  -\-  2  tan  x  an  odd  or  an  even  function?  Is 
sin  X  +  cos  X  an  odd  or  an  even  function  of  x? 

60.  The  Defining  Equations  cleared  of  Fractions.     The  student 
should  commit  to  memory  the  equations  defining  the  trigonometric 
functions  when  cleared  of  fractions.     In  this  form  the  equations 
are  quite  as  useful  as  the  original  ratios.     They  are  written: 
y  =  a  sin  ^  x  =  y  cot  ^ 

X  =  SL  cos  d  a  =  X  sec  ^ 

y  =  X  tan  0  a  =  y  esc  ^ 

As  applied  to  the  right  angled  triangle,  they  may  be  stated  in 
words  as  follows: 

Either  leg  of  a  right  triangle  is  equal  to  the  hypotenuse  multiplied 
by  the  sine  of  the  opposite,  or  by  the  cosine  of  the  adjacent,  angle. 

Either  leg  of  a  right  triangle  is  equal  to  the  other  leg  multiplied  by  the 
tangent  of  the  opposite,  or  by  the  cotangent  of  the  adjacent,  angle. 


§61]    THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS    117 

The  hypotenuse  of  a  right  triangle  is  equal  to  either  leg  multiplied 
by  the  secant  of  the  angle  adjacent,  or  by  the  cosecant  of  the  angle 
opposite  that  leg. 

These  statements  should  be  committed  to  memory. 

61.  Projections.  In  Fig.  52  the  projection  of  OP  in  any  of  its 
positions,  such  as  OPi,  OP2,  OP 3,  .  .  .  ,  is  ODi,  ODz,  OD3,  .  .  .  , 
or  is  the  abscissa  of  the  point  P.     Thus  for  all  positions: 

X  =  a  cos  6 
The  sign  of  x  gives  the  sign,  or  sense,  of  the  projection.     In  each 
case  6  is  said  to  be  the  angle  of  projection. 

The  above  definition  of  projection  is  more  general  in  one 
respect  than  that  discussed  in  §28.  By  the  present  definition 
the  projection  of  a  line  is  negative  if  90°  <  ^  <  270°  (read, 
"if  d  is  greater  than  90°  but  is  less  than  270°").  This  con- 
cept is  important  and  essential  in  expressing  a  component  of  a 
displacement,  of  a  velocity,  of  an  acceleration,  or  of  a  force. 

The  cosine  of  d  might  have  been  defined  as  that  proper  fraction 
by  which  it  is  necessary  to  multiply  the  length  of  a  fine  in  order  to 
produce  the  projection  of  the  line  on  a  line  making  an  angle  6 
with  it. 

Exercises 

1.  A  stretched  guy  rope  makes  an  angle  of  60°  with  the  horizontal. 
What  is  the  projection  of  the  rope  on  a  horizontal  plane?  What  is 
the  projection  of  the  rope  on  a  vertical  plane? 

2.  Find  the  lengths  of  the  projections  of  the  line  through  the  origin 
and  the  point  (1,  VS)  upon  the  OX  and  OY  axes,  if  the  line  is  12 
inches  long. 

3.  A  force  equals  200  dynes.  What  is  its  component  (projection) 
on  a  line  making  an  angle  of  135°  with  the  force?  On  a  line  making 
an  angle  of  120°  with  the  force? 

4.  A  velocity  of  20  feet  per  second  is  represented  as  the  diagonal 
of  a  rectangle  the  longer  side  of  which  makes  an  angle  of  30°  with  the 
diagonal.  Find  the  components  of  the  velocity  along  each  side  of  the 
rectangle. 

6.  Show  that  the  projections  of  a  fixed  line  OA  upon  all  other 
lines  drawn  through  the  point  0  are  chords  of  a  circle  of  diameter  OA . 
See  Fig.  63. 

6.  Find  the  projection  of  the  side  of  a  regular  hexagon  upon  the 
three  diagonals  passing  through  one  end  of  the  given  side,  if  the 


118        ELEMENTARY  MATHEMATICAL  ANALYSLS        [§02 


numerical  value  of   cos  30°  =  0.87,  and  if  each  side  of  the  hexagon 
is  20  feet. 

62.  Polar  Coordinates.  In  Fig.  61,  the  position  of  the  point 
P  may  be  assigned  either  by  giving  the  x  and  y  of  the  rectangular 
coordinate  system,  or  by  giving  the  vectorial  angle  6  and  the 
distance  OP  measured  along  the  terminal  side  of  d.  Unlike 
the  distance  a  used  in  the  preceding  work,  it  is  found  conven- 
ient to  give  the  line  OP  a  sense  or  direction  as  well  as  length  ; 
such  a  line  is  called  a  vector.     In  the  present  case,  it  is  known  as 

the  radius  vector  of  the  point  P, 
and  it  is  usually  symbolized  by 
the  letter  p.  The  vectorial  or 
direction  angle  6  and  the  radius 
vector  p  are  together  called  the 
polar  coordinates  of  the  point  P, 
and  the  method,  as  a  whole,  is 
known  as  the  system  of  polar 
coordinates.  In  Fig.  61  the 
point  P'  is  located  by  turning 
from  the  fundamental  direction 
OX,  called  the  polar  axis,  through 
an  angle  6  and  then  stepping 
backward  the  distance  p  to  the 
point  P';  this  is,  then,  the  point  {  —  p,  6).  P'  has  also  the  coordi- 
nates (p,  0^,  in  which  62  —  6  +  180°;  likewise  Pi  is  (+  p',  61)  and 
P'l  is  (—  p',  ^1).  Thus  each  point  may  be  located  in  the  polar 
system  of  coordinates  in  two  ways,  i.e.,  with  either  a  positive  or  a 
negative  radius  vector.  If  negative  values  of  6  be  used,  there 
are  Jour  ways  of  locating  a  point  without  using  values  of  ^  > 
360°.  In  giving  a  point  in  polar  coordinates,  it  is  usual  to  name 
the  radius  vector  first  and  then  the  vectorial  angle;  thus  (5,  40°) 
means  the  point  of  radius  vector  5  and  vectorial  angle  40°. 

63.  Polar  Coordinate  Paper.  Polar  coordinate  paper  (form  MS) 
is  prepared  for  the  construction  of  loci  in  the  polar  system.  A  re- 
duced copy  of  a  sheet  of  such  paper  is  shown  in  Fig.  62.  This 
plate  is  graduated  in  degrees,  but  a  scale  of  radian  measure  is  given 
in  the  margin.  The  radii  proceeding  from  the  pole  0  meet  all  of 
the  circles  at  right  angles,  just  as  the  two  systems  of  straight  lines 


Fig.  G1. — Polar  Coordinates. 


§03]    THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS    119 


meet  each  other  at  right  angles  in  rectangular  coordinate  paper. 
For  this  reason,  both  the  rectangular  and  the  polar  systems  are 
called  orthogonal  systems  of  coordinates. 

We  have  learned  that  the  fundamental  notion  of  a  function 
implies  a  table  of  corresponding  values  for  two  variables,  one  called 
the  argument  and  the  other  the  function.     The  notion  of  a  graph 


Fig.  62. — Polar  Cooradinate  Squared  Paper.     (From  M3.) 

implies  any  sort  of  a  scheme  for  a  pictorial  representation  of  this 
table  of  values.  There  are  three  common  methods  in  use :  the  double 
scale,  the  rectangular  coordinate  paper,  and  the  polar  paper.  The 
polar  paper  is  most  convenient  in  case  the  argument  is  an  angle 
measured  in  degrees  or  in  radians.  Since  in  a  table  of  values  for  a 
functional  relation  we  need  to  consider  both  positive  and  negative 
values  for  both  the  argument  and  the  function,  it  is  necessary  to 
use  on  the  polar  paper  the  convention  already  explained.  The 
argument,  which  is  the  angle,  is  measured  counter-clockwise  if 
positive  and  clockwise  if  negative  from  the  line  numbered  0°, 


120        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§(54 

Fig.  62.  The  function  is  measured  outward  from  the  center  along 
the  terminal  side  of  the  angle  for  positive  functional  values  and 
outward  from  the  center  along  the  terminal  side  of  the  angle 
produced  backward  through  the  center  for  negative  functional  values. 
In  this  scheme  it  appears  that  four  different  pairs  of  values  are 
represented  by  the  same  point.  This  is  made  clear  by  the  points 
plotted  in  the  figure.      The  points  Pi,  P2,  Pz,  Pi  are  as  follows: 

Pi  :  (6.0,40°);  (6.0,  -  320°);  (-  6.0,220°);  (-  6.0,  -  140°). 
P2  :  (10,  135°);  (10,   -  225°);  (-  10,  315°);  (-  10,-  45°). 
P3  :  (5,  230°);  (5,  -  130°);  (-  5,  50°);  (-  5,  -  310°). 
P4  :  (6.0,330°);  (6.0,  -  30°);  (-  6.0,  150°);  (-  6.0,  -  210°). 

The  angular  scale  cannot  be  changed,  but  the  functional  scale 
can  be  changed  to  suit  the  table  of  values  by  multiplying  or 
dividing  it  by  integral  powers  of  ten. 

In  case  the  vectorial  angle  is  given  in  radians,  the  point  may  be 
located  on  the  polar  paper  by  means  of  a  straight  edge  and  the 
marginal  scale  on  form  M3. 


-^^v-- 


Exercises 


1.  Locate  the  following  points  on  polar  coordinate  paper;  (1,  x/2); 
(2,  x);  (3,  60°);  (4,  250°);  (2^  I.Stt). 

2.  Locate  the  following  points:  (0,  0°);  (1,  10°);  (2,  20°);  (3,  30°); 
(4,  40°);  (5,  50°);  (6,  60°);  (7,  70°);  .  .  .  (36,  360°).  Use  1  cm.  =  10 
units. 

3.  The  equation  of  a  curve  in  polar  coordinates  is  0  =  2.  Draw 
the  curve.  The  equation  of  a  second  curve  is  p  =3.  Draw  the 
curve. 

Notice  that  p  =  a  constant  is  a  circle  with  center  at  0,  while 
6  =  a.  constant  is  a  straight  line  through  0. 

4.  Draw  the  curve  p  =  6  using  2  cm.  as  unit  for  p.  Note  that  the 
curve  p  =  0  is  a  spiral  while  the  curve  y  =  a;  is  a  straight  line. 

64.  Graphs  of  p  =  a  cos  6  and  p  =  a  sin  6.  These  are  two 
fundamental  graphs  in  polar  coordinates.  The  equation 
p  =  a  cos  d  states  that  p  is  the  projection  of  the  fixed  length  a 
upon  a  radial  line  proceeding  from  0  making  a  direction  angle  6 
with  a,  or,  in  other  words,  p  in  all  of  its  positions  must  be  the  side 
adjacent  to  the  direction  angle  ^  in  a  right  triangle  whose  hypote- 
nuse is  the  finite  length  a.     (See  §61.)     It  must  be  remem- 


§64]    THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS    121 

bered  that  the  direction  angle  0  is  always  measured  from  the  fixed 
direction  OA.  Hence,  to  construct  the  locus  p  =  a  cos  6,  draw 
as  many  radii  vectores  as  desired,  as  in  Fig.  63.  Project  on  each 
of  these  the  fixed  distance  OA  or  a.  This  gives  OP,  or  p,  in  numer- 
ous positions  as  shown  in  the  diagram.  Since  P  is  by  construction 
the  foot  of  the  perpendicular  dropped  from  A  upon  OP,  it  is  always 
at  the  vertex  of  a  right  triangle  standing  on  the  fixed  hypotenuse  a, 
and  therefore  the  point  P  is  on  the  semicircle  AOP;  for,  from  plane 
geometry  a  right  triangle  is  always  inscribable  in  a  semicircle. 


Fig.  63. — The  Graph  of  p  =  a  cos  d. 


When  0  is  in  the  second  quadrant,  as  02,  Fig.  63,  the  cosine  is 
negative  and  consequently  p  is  also  negative.  Therefore  the  point 
P2  is  located  by  measuring  backward  through  0.  Since,  however, 
P2  is  the  projection'  of  a  through  the  angle  62  (see  §61),  the 
angle  at  P2  must  be  a  right  angle.  Thus  the  semicircle 
OP2A  is  described  as  6  sweeps  the  second  quadrant.  When 
6  is  in  the  third  quadrant,  as  ^3,  the  cosine  is  still  negative  and 
p  is  measured  backward  to  describe  the  semicircle  APiO  a  second 
time.  As  6  sweeps  the  fourth  quadrant,  the  semicircle  OP2A  is 
described  the  second  time.  Thus  the  graph  in  polar  coordinates 
of  p  =  a  cos  6  is  a  circle  twice  drawn  as  6  varies  from  0°  to  360°. 
Once  around  the  circle  corresponds  to  the  distance  from  crest  to 
trough  of  the  "wave"  y  =  a  cos  x,  in  Fig.  59  {0'  is  origin). 
The  second  time  around  the  circle  corresponds  to  the  distance 


122        ELEMENTARY  MATHEMATICAL  ANALYSIS 


^64 


from  trough  to  crest  of  the  cosine  curve.  Trough  and  crest  of  all 
the  successive  "  wave  lengths  "  fall  at  the  point  A .  The  nodes  are 
all  at  0. 

The  polar  representation  of  the  cosine  of  a  variable  by  means 
of  the  circle  is  more  useful  and  important  in  science  than  the 
Cartesian  representation  by  means  of  the  sinusoid.  The  ideas 
here  presented  must  be  thoroughly  mastered  by  the  student. 

The  graph  oi  p  =  a  sin  6  is  also  a  circle,  but  the  diameter  is 
the  line  OB  making  an  angle  of  90°  with  OA,  as  shown  in  Fig.  64. 

Since  p  =  a  sin  6,  the  radius  vector, 
as  6  increases  to  90°,  must  equal 
the  side  lying  opposite  the  angle  d 
in  a  right  triangle  of  hypotenuse  a. 
Since  angle  AOPi  =  angle  OBPi, 
the  point  P  may  be  the  vertex  of 
any  right  triangle  erected  on  OB  or 
a  as  a  hypotenuse.  The  semicircle 
BP2O  is  described  as  6  increases 
from  90°  to  180°.  Beyond  180°  the 
sine  is  negative,  so  that  the  radius 
vector  p  must  be  laid  off  backward 
for  such  angles.  Thus  P3  is  the 
point  corresponding  to  the  angle  ^3,  of  the  third  quadrant.  As  0 
sweeps  the  third  and  fourth  quadrants  the  circle  OP1BP2O  is 
described  a  second  time.  Therefore,  the  graph  of  p  =  a  sin  ^ 
is  the  circle  twice  drawn  of  diameter  a,  and  tangent  to  OX  at  0. 
The  first  time  around  the  circle  corresponds  to  the  crest,  the 
second  time  around  corresponds  to  the  trough  of  the  wave  or 
sinusoid  drawn  in  rectangular  coordinates.  The  points  corre- 
sponding to  the  nodes  of  the  sinusoid  are  at  0  and  the  points 
corresponding  to  the  maximum  and  minimum  points  are  at  B. 

We  have  seen  that  the  graph  of  a  function  in  polar  coordinates 
is  a  very  different  curve  from  its  graph  in  rectangular  coordi- 
nates. Thus  the  cosine  of  a  variable  if  graphed  in  rectangular 
coordinates  is  a  sinusoid;  but  if  graphed  in  polar  coordi- 
nates the  graph  is  a  circle  (twice  drawn).  There  is  in  this  case 
a  very  great  difference  in  the  ease  with  which  these  curves  can  be 
constructed;  the  sinusoid  requires  an  elaborate  method,  while  the 


Fig 


64. — The  Graph  of  p  =  a 
sin  6. 


§65]    THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS    123 

circle  may  be  drawn  at  once  with  compasses.  This  is  one  reason 
why  the  periodic  or  sinusoidal  relation  is  preferably  represented 
in  the  natural  sciences  by  polar  coordinates. 

65.  Graphical  Table  of  Sines  and  Cosines.  The  polar  graphs 
of  p  =  a  sin  ^  and  p  =  a  cos  ^  furnish  the  best  means  of  construct- 
ing graphical  tables  of  sines  and  cosines.  The  two  circles  passing 
through  0  shown  on  the  polar  coordinate  paper,  form  M3,  Fig.  62, 
are  drawn  for  this  purpose.  A  quantity  of  this  coordinate  paper 
should  be  in  the  hands  of  the  student.  If  the  diameter  of  the 
sine  and  cosine  circles  be  called  1,  then  the  radius  vector  of  any 
point  on  the  lower  circle  is  the  cosine  of  the  vectorial  angle,  and 
the  radius  vector  of  the  corresponding  point  on  the  upper  circle 
is  the  sine  of  the  vectorial  angle.  As  there  are  50  concentric  cir- 
cles in  Form  MS,  it  is  easy  to  read  the  radius  vector  of  a  point 
to  1  /lOO  of  the  unit.  Thus,  from  the  diagram,  we  read  cos 
45°  =  0.70 ;  cos  60°  =  0.50 ;  cos  30°  =  0.866.  These  results  are 
nearly  correct  to  the  third  place. 

66.  Graphical  Table  of  Tangents  and  Secants.  Referring  to 
Fig.  62,  it  is  obvious  that  the  numerical  values  of  the  tangents  of 
angles  can  be  read  off  by  use  of  the  uniform  scale  of  centimeters 
bordering  the  polar  paper  (form  MS).  The  scale  referred  to 
lies  just  inside  of  the  scale  of  radian  measure,  and  is  numbered 
0,  2,  4,  .  .  . ,  at  the  right  of  Fig.  62.  Thus  to  get  the  numerical 
value  of  tan  40°  it  is  merely  necessary  to  call  unity  the  side  OA 
of  the  triangle  of  reference  OAP,  and  then  read  the  side  AP  =  0.84; 
hence  tan  40°  =  0.84.  To  the  same  scale  (i.e.,  OA  =  1)  the  dis- 
tance OP  =  1.31,  but  this  is  the  secant  of  the  angle  AOP,  whence 
sec  40°  =  1.31.  By  use  of  the  circles  we  find  sin  40°  =  0.64  and 
cos  40°  =  0.76. 

In  case  we  are  given  an  angle  greater  than  45°  (but  less  than 
135°)  use  the  horizontal  scale  through  B.  Starting  from  B  as 
zero  the  distance  measured  on  the  horizontal  scale  is  the  cotangent 
of  the  given  angle.  The  tangent  is  found  by  taking  the  reciprocal 
of  the  cotangent. 

Exercises 

Find  the  unknown  sides  and  angles  in  the  following  right  triangles. 
The  numerical  values  of  the  trigonometric  functions  are  to  be  taken 


124        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§66 

from  the  polar  paper.  The  vertices  of  the  triangles  are  supposed 
to  be  lettered  A,  B,  C  with  C  at  the  vertex  of  the  right  angle.  The 
small  letters  a,  h,  c  represent  the  sides  opposite  the  angles  of  the  same 
name. 

By  angle  of  elevation  is  meant  the  angle  between  a  horizontal  line 
and  a  line  to  the  object,  both  drawn  from  the  point  of  observation, 
when  the  object  lies  above  the  horizontal  line.  The  similar  angle 
when  the  object  lies  helow  the  observer  is  called  the  angle  of  depression 
of  the  object. 

The  solution  of  each  of  the  following  problems  must  be  checked. 
The  easiest  check  is  to  draw  the  triangles  accurately  to  scale  on  form 
Ml  and  use  a  protractor. 

1.  When  the  altitude  of  the  sun  is  40°,  the  length  of  the  shadow  cast 
by  a  flag  pole  on  a  horizontal  plane  is  90  feet.  Find  the  height  of  the 
pole. 

Outline  of  Solution.  Call  height  of  pole  a,  and  length  of  shadow  h. 
Then  A  =  40°  and  B  =  50°.     Hence: 

a  =  6  tan  40° 

Determining  the  numerical  value  of  the  tangent  from  the  polar  paper, 
we  find: 

a  =  90  X  0.84  =  75.6  ft. 

which  result,  if  checked,  is  the  height  of  the  pole.  To  check,  either 
draw  a  figure  to  scale,  or  compute  the  hypotenuse  c,  thus: 

c  =  90  sec  40° 
From  the  polar  paper  find  sec  40°.     Then: 

c  =  90  X  1.31  =  117.9 
Since  a^  +  b^  =  c^,  we  have  c^  ~  b^  =  a^,  or  (c  -  6)  (c  +  6)  =  a^. 
Hence  if  the  result  found  be  correct, 

(117.9  -  90)  (117.9  +  90)  =  75.62 
5800  =  5715 

These  results  show  that  the  work  is  correct  to  about  three  figures,  for 
the  sides  of  the  triangle  are  proportional  to  the  square  roots  of  the 
numbers  last  given. 

2.  At  a  point  200  feet  from,  and  on  a  level  with,  the  base  of  a  tower 
the  angle  of  elevation  of  the  top  of  the  tower  is  observed  to  be  60°. 
What  is  the  height  of  the  tower? 

3.  A  ladder  40  feet  long  stands  against  a  building  with  the  foot  of 
the  ladder  15  feet  from  the  base  of  the  wall.  How  high  does  the 
ladder  reach  on  the  wall? 


§66]    THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS   125 

4.  From  the  top  of  a  vertical  cliff  the  angle  of  depression  of  a  point 
on  the  shore  150  feet  from  the  base  of  the  cliff  is  observed  to  be  30°. 
Find  the  height  of  the  cliff. 

5.  In  walking  half  a  mile  up  a  hill,  a  man  rises  300  feet.  Find  the 
angle  at  which  the  hill  slopes. 

If  the  hill  does  not  slope  uniformly  the  result  is  the  average  slope 
of  the  hill. 

6.  A  line  3.5  inches  long  makes  an  angle  of  35°  with  OX.  Find  the 
lengths  of  its  projections  upon  both  OX  and  OY. 

7.  A  vertical  cliff  is  425  feet  high.  From  the  top  of  the  cliff  the 
angle  of  depression  of  a  boat  at  sea  is  16°.  How  far  is  the  boat 
from  the  foot  of  the  cliff? 

8.  The  projection  of  a  line  on  OX  is  7.5  inches,  and  its  projection 
on  OY  is  1.25  inches.  Find  the  length  of  the  line,  and  the  angle 
it  makes  with  OX. 

9.  A  battery  is  placed  on  a  cliff  5 10  feet  high.  The  angle  of  depres- 
sion of  a  floating  target  at  sea  is  9°.  Find  the  range,  or  the  distance 
of  the  target  from  the  battery. 

10.  From  a  point  A  the  angle  of  elevation  of  the  top  of  a  monument 
is  25°.  From  the  point  B,  110  feet  farther  away  from  the  base  of  the 
monument  and  in  the  same  horizontal  straight  line,  the  angle  of  eleva- 
tion is  15°.     Find  the  height  of  the  monument. 

11.  Find  the  length  of  a  side  of  a  regular  pentagon  inscribed  in  a 
circle  whose  radius  is  12  feet. 

12.  Proceeding  south  on  a  north  and  south  road,  the  direction  of  a 
church  tower,  as  seen  from  a  milestone,  is  41°  west  of  south.  From 
the  next  milestone  the  tower  is  seen  at  an  angle  of  65°  W.  of  S. 
Find  the  shortest  distance  of  the  tower  from  the  road. 

13.  A  traveler's  rule  for  determining  the  distance  one  can  see  from 
a  given  height  above  a  level  surface  (such  as  a  plain  or  the  sea)  is  as 
follows :  "  To  the  height  in  feet  add  half  the  height  and  take  the  square 
root.  The  result  is  the  distance  you  can  see  in  miles."  Show  that 
this  rule  is  approximately  correct,  assuming  the  earth  a  sphere  of 
radius  3960  miles.  Show  that  the  drop  in  1  mile  is  8  inches,  and 
that  the  water  in  the  middle  of  a  lake  8  miles  in  width  stands  10|  feet 
higher  than  the  water  at  the  shores. 

14.  Observations  of  the  height  of  a  mountain  were  taken  at  A  and 
B  on  the  same  horizontal  line  and  in  the  same  vertical  plane  with  the 
top  of  the  mountain.  The  elevation  of  the  top  at  A  is  52°  and  at  jB  is 
36°.     The  distance  AB  is  3500  feet.     Find  the  height  of  the  mountain. 

16.  The  diagonals  of  a  rhombus  are  16  and  20  feet,  respectively. 
Find  the  lengths  of  the  sides  and  the  angles  of  the  rhombus. 


126 


ELEMENTARY  MATHEMATICAL  ANALYSIS 


[§67 


125- 


FiG.  65. — Diagram  for 
Exercise  17. 


16.  The  equation  of  a  line  is  y  =  f a:  +  10.  Compute  the  short- 
est distance  of  this  Hne  from  the  origin. 

17.  Find  the  perimeter  and  area  of  ABCD,  Fig.  65. 

67.  The  Law  of  the  Circular  Functions.  It  will  be  emphasized 
in  this  book  that  the  fundamental  laws  of  exact  science  are  three  in 
number,  namely:  (1)  The  power  function 
expressed  by  y  =  ax"  where  n  may  be 
either  positive  or  negative;  (2)  the  har- 
monic or  periodic  law  y  =  a  sin  nx,  which 
is  fundamental  to  all  periodically  occurring 
phenomena;  and  a  third  law  to  be  dis- 
cussed in  a  subsequent  chapter.  While 
other  important  laws  and  functions  arise 
in  the  exact  sciences,  they  are  secondary 
to  those  expressed  by  the  three  funda- 
mental relations. 

We  have  stated  the  law  of  the  power 

function  in  the  following  words  (see  §34) : 

hi  any  power  function,  if  x  change  by  a  fixed  multiple,  y  is 

changed  by  a  fixed  multiple  also.     In  other  words,  if  x  change  by 

a  constant  factor,  y  will  change  by  a  constant  factor  also. 

Confining  our  attention  to  the  fundamental  functions,  sine 
and  cosine,  in  terms  of  which  the  other  circular  functions  can 
be  expressed,  we  may  state  their  law  as  follows:^ 

The  circular  functions,  sin  6  and  cos  6,  change  periodically  in 
value  proportionally  to  the  periodic  change  in  the  ordinate  and 
abscissa,  respectively,  of  a  point  moving  uniformly  on  the  circle 
:c2  4-  2/2  =  a2. 

The  use  of  the  periodic  law  in  natural  science  is,  of  course, 
very  different  from  that  of  the  power  function.  The  student  will 
find  that  circular  functions  similar  toy  =  a  sin  nx  wiU  be  required 
in  order  to  express  properly  any  phenomena  which  are  recurrent 
or  periodic  in  character,  such  as  the  motion  of  vibrating  bodies, 
all  forms  of  wave  motion,  such  as  sound  waves,  light  waves,  electric 
waves,  alternating  currents  and  waves  on  water  surfaces,  etc. 
Almost  every  part  of  a  machine,  no  matter  how  complicated  its 
motions,   repeats  the  original  positions  of  all  of  the  parts  at 

^  Chapter  X  is  devoted  to  a  discussion  of  these  fundamental  periodic  laws. 


§68]    THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS    127 

stated  intervals  and  these  recurrent  positions  are  expressible  in 
terms  of  the  circular  functions  and  not  otherwise.  The  student 
will  obtain  a  most  limited  and  unprofitable  idea  of  the  use  of  the 
circular  functions  if  he  deems  that  their  principal  use  is  in  numer- 
ical work  in  solving  triangles,  etc.  The  importance  of  the 
circular  functions  lies  in  the  power  they  possess  of  expressing 
natural  laws  of  a  periodic  character. 

68.  Rotation  of  Any  Locus.  In  §36  we  have  shown  that 
any  locus  y  =  f{x)  is  translated  a  distance  a  in  the  x  direction  by 
substituting  {x  —  a)  for  x  in  the  equation  of  the  locus.  Likewise 
the  substitution  of  (?/  —  b)  for  y  was  found  to  translate  the  locus 
the  distance  b  in  the  y  direction.  A  discussion  of  the  rotation  of 
a  locus  was  not  considered  at  that  place,  because  a  displacement 
of  this  type  is  best  brought  about  when  the  equations  are  expressed 
in  polar  coordinates. 

If  a  table  of  values  be  prepared  for  each  of  the  loci 

p  =  cos  6  (1) 

p  =  cos  (^1  -  30°)  .  (2) 

as  follows: 

e  !  0°         30°         60°       90°       120°  150°  180°     , 


P  I    1        WS       1/2         0        -1/2     -  i\/3         -1 
di     I    30°  60°         90°       120°    ■   150°  180° 


p     I      1  |\/3       1/2         0        -1/2     -  i\/3        . 

and  then  if  the  graph  of  each  be  drawn,  it  will  be  seen  that  the 
curves  differ  only  in  their  location  and  not  at  all  in  shape  or  size. 
The  reason  for  this  is  obvious:  The  same  value  of  p  is  given  by 
^1  =  90°  in  the  second  case  as  is  given  by  ^  =  60°  in  the  first 
case,  and  the  same  value  of  p  is  given  by  ^i  =  60°  in  the  second 
case  as  is  given  by  ^  =  30°  in  the  first  case,  etc.  The  sets 
of  values  of  p  in  the  two  cases  are  identical,  but  like  values  corre- 
spond to  vectorial  angles  0  differing  by  30°.  In  more  general 
terms  the  reasoning  is  that  if  (0i  -^  30°)  be  substituted  for  6  in  any 
polar  equation,  then  since  (^i  —  30°)  has  been  put  equal  to  6,  it 
follows  that  Oi  =  {6  -\-  30°),  or  the  new  vectorial  angle  ^i  is  greater 
than  the  original  6  by  the  amount  30°.     Since  all  values  of  0 


128        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§68 


in  the  new  locus  are  increased  by  30°,  the  new  locus  is  the 
same  as  the  original  locus  rotated  about  0  (positive  rotation) 
by  the  amount  30°. 

The  above  reasoning  does  not  depend  upon  the  particular  con- 
stant angle  30°  that  happened  to  be  used,  but  holds  just  as  well 
if  any  other  constant  angle,  say  a,  be  used  instead.  That  is, 
substituting  (^i  —  a)  for  6  does  not  change  the  size  or  shape  of 
the  locus,  but  merely  rotates  it  through  an  angle  a  in  the  positive 

sense.  The  same  reasoning 
applies  also  to  the  general 
case :  If  p  =  f{d)  be  the  polar 
equation  of  any  locus,  then  p 
=  /(^i  ~  <^)  is  the  equation 
of  the  same  locus  turned 
about  the  fixed  point  0 
through  the  angle  a;  for  if 
(01  —  a)  be  everywhere  substi- 
tuted for  the  vectorial  angle 
6,  01  must  be  a  greater  than 
the  old  0.  That  is,  each 
point  is  advanced  the  angular 
amount  a,  or  turned  that 
amount  about  the  point  0. 
The  rotation  is  positive,  or 
anti-clockwise,  if  a  be  posi- 
tive— thus,  substituting  (0  —  30°)  for  ^  in  p  =  a  cos  0  turns  the 
circle  p  =  a  cos  ^  through  30°  in  the  anti-clockwise  sense,  as  is 
shown  in  Fig.  66,  but  substituting  (0  -\-  30°)  for  ^  in  p  =  a  cos  0 
turns  the  circle  p  =  a  cos  ^  through  30°  in  the  clockwise  direc- 
tion of  rotation,  as  shown  in  the  same  figure. 
The  four  circles 


Fig.  66. — Rotation  of  the  Circles  p 
a  cos  6  and  p  =  a  sin-0. 


are  shown  in  Fig. 


p  =  a  cos  (0  -f-  a) 
p  =  a  cos  (0  —  a) 
p  =  a  sin  (^  -j-  a) 
p  =  a  sin  (0  —  a) 
Each  has  diameter  a. 


(3) 
(4) 
(5) 
(6) 
The  student  must 


carefully  distinguish  between  the  constant  angle  a  and  the  variable 


§69]    THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS   129 

angle  6,  just  as  he  must  distinguish  between  the  constant  distance 
a  and  the  variable  vector  p. 

The  above  result  constitutes  another  of  the 

Theorems  on  Loci 

IX.  If  {d  —  a)  be  substituted  for  6  throughout  the  polar  equation 
of  any  locus,  the  curve  is  rotated  through  the  angle  a  in  the  positive 
sense. 

Note  that  the  substitution  is  (^  —  a)  for  6  when  the  required 
rotation  is  through  the  positive  angle  a,  and  that  the  substitution 
is  (0  +  a)  for  6  when  the  required  rotation  is  through  the  negative 
angle  a. 

The  rotation  of  any  locus  through  any  angle  is  readily  accom- 
plished when  its  equation  is  given  in  polar  coordinates.  Rota- 
tions of  180°  and  90°  are  very  simple  in  rectangular  coordinates. 
Let  the  student  select  any  point  P  in  rectangular  coordinates  and 
draw  the  radius  vector  OP  and  the  abscissa  and  ordinate  OD  and 
DP;  then  show  that  the  substitutions  x  =  —xi,y=  —yi  will  turn 
OP  through  180°  about  0  in  the  plane  a;?/,  and  that  the  substitutions 
X  =  yi,y  =  —  xi  will  turn  OP  through  90°  about  0  in  the  plane  xy. 

Exercises 
Draw  the  following  circles : 

1.  p  =  3  cos  (d  -  30)^  4.  p  =  2  sin  (d  +  135°). 

2.  p  =  3  cos  (d  +  120°).  5.  p  =  4  cos  (d  +  | )  • 

3.  p  =  2  sin  (d  -  45°).  6.  p  =  5  sin  f|  -  d))  • 

7.  Show  that  p  =  a  sin  d  is  the  locus  p=  a  cos  d  rotated  90° 
counter  clockwise. 

Solution:    Write  p=  a  cos  (0-90°),  then 

p=a  cos  (90° -0)  by  (2)  §68,  then  p=  a  sin  0  by  §57. 

69.  Polar  Equation  of  the  Straight  Line.  In  Fig.  67  let  MN  be 
any  straight  line  in  the  plane  and  07"  be  the  perpendicular  dropped 
upon  MN  from  the  origin  0.  Let  the  length  of  07"  be  a  and  let 
the  direction  angle  of  OT  be  a,  "where,  for  a  given  straight  line, 
a  and  a  are  constants.    Let  p  be  the  radius  vector  of  any  point 

9 


130        ELEMENTARY  MATHEMATICAL  ANALYSIS         [§69 


P  on  the  line  MN  and  let  its  direction  angle  be  6.     Then,  by 
definition, 

-  =  cos  id  —  a) 
P 

Therefore  the  equation  of  the  straight  line  MN  is 

a  =  p  cos  {6  —  a)  (1) 

for  it  is  the  equation  satisfied  by  the  (p,  6)  of  every  point  of  the 
line.     This  is  the  equation  of  any  straight  line,  for  its  location  is 

perfectly  general.  The 
constants  defining  the  line 
are  the  perpendicular  dis- 
tance a  upon  the  given  line 
from  0  and  the  direction 
angle  a  of  this  perpendic- 
ular. The  perpendicular 
OT  or  a  is  called  the  nor- 
mal to  the  line  MN  and 
the  equation  (1)  is  called 
the  normal  equation  of  the 
straight  line. 

The  equation  of  the  cir- 
cle shown  in  the  figure  is 

pi  =  a  cos  {6  —  a)     (2) 
in  which  pi  represents  the 
radius  vector  of  a  point  Pi 
on  the  circle.     From  plane  geometry  OT  or  a  is  a  mean  propor- 
tional between  the  secant  OP  and  the  chord  0P\,  or, 


Fig.  67.— The  Circle  p  = 
and  its  Inverse,  the  line 
p  cos  (,d  —  a). 


a  cos  {6  — 
MN  or  a 


«) 


p\a  =  a:  pi 


or. 


PPi  =  a2  (3) 

This  gives  the  relation  between  the  radius  vector  of  a  point  on  the 
line  and  the  corresponding  radius  vector  of  a  point  on  the  circle. 
Now  if  on  the  radius  vector  p  =  OP,  drawn  from  the  fixed  origin 


=       (where  a  is  a 


0  to  any  curve,  we  lay  off  a  length  OPi  =  pi 

constant),  then  Pi  is  said  to  describe  the  inverse  of  the  given  curve 
with  respect  to  0,     In  this  special  case  the  circle  is  the  inverse  of 


§70]    THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS    131 

the  straight  line  and  vice  versa.  If  a  =  1  we  note  that  OPi  and 
OP  are  reciprocals  of  each  other. 

It  is  important  in  mathematics  to  associate  the  equation  of  the 
circle  and  the  equation  of  its  inverse  with  respect  to  0,  or  the  line 
tangent  to  it.     Thus 

p  =  10  cos  (d  -  ^) 
is  a  circle 

10  =  p  cos  Id  -  j) 

is  a  straight  line  tangent  to  it. 
70.  Relation  between   Rectangular   and   Polar   Coordinates. 

Think  of  the  point  P  whose  rectangular  coordinates  are  {x,  y). 
If  the  radius  vector  OP  be  called  p  and  the  direction  angle  be 
called  6,  then  the  polar  coordinates  of  P  are  (p,  B) .  Then  x  and 
y  for  any  position  of  P  are  the  projections  of  p  through  the 
angle  6,  and  the  angle  (90°  —  ^),  respectively,  or, 

X  =  p  cos  B  (1) 

y  =  p  sin  ^  (2) 

These  are  the  equations  of  transformation  that  permit  us  to  express 
the  equation  of  a  curve  in  polar  coordinates  when  its  equation  in 
rectangular  coordinates  is  known,  and  vice  versa.  Thus  the 
straight  line  x  =  3  has  the  equation 

p  cos  ^  =  3 

in  polar  coordinates.     The  line  x  -{-  y  =  3  has  the  polar  equation 

p  cos  ^  +  p  sin  ^  =  3. 

The  circle  x^  -\-  y"^  =  a^  has  the  equation 

p2-cos2  0  +  p2sin2(9  =  a2 
or, 

p2  =  a2 

or,      , 

p  =  a 
etc. 


132        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§71 
To  solve  equations  (1)  and  (2)  for  B,  we  write 

X 

d  =  the  angle  whose  cosine  is  ^ 

P 

y 

0  =  the  angle  whose  sine  is  - 

P 

The  verbal  expression  "the  angle  whose  cosine  is,"   etc.,   are 

abbreviated    in    mathematics   by  the  notations   "cos~^,"    read 

*' anti-cosine,"  and  "sin-V'  read  "anti-sine,"  as  follows: 

e  =  cos-i  (x/p)  (3) 

d  =  sin-i  (y/p)  (4) 

Dividing  the  members  of  (2)  by  the  members  of  (1)  we  obtain 

y 

tan  6  =     '  which,  solved  for  6,  we  write 

X  '  ' 

y 

6  =  the  angle  whose  tangent  is  — 

X 

which  may  be  abbreviated 

6  =  tan-i  (y/x)  (5) 

and  read  "6  =  the  anti- tangent  of  2//x." 
The  value  of  p  in  terms  of  x  and  y  is  readily  written 


p  =  \/x2  -h  y2  (6) 

Exercises 

1.  Write  in  polar  coordinates  the  equation  x^  -\-  y^  +  8x  =  0. 
The  result  is  p^  +  8p  cos  0  =  0,  or  p  =  —8  cos  6. 

2.  Write  in  polar  coordinates  the  equations  (a)  x^  -\-  y^  —  4y  =  0; 
(6)  a;2  -h  2/2  -  6x  -  4^  =  0;  (c)  x^  +  y^  -  Qy  =  4. 

3.  Write  in  polar  coordinates  the  equations  (a)  x  +?/  =  l;  ih)  x-\-2y 
=  l]{c)  X  +  VZy  =  2. 

4.  Write  in  rectangular  coordinates  (a)  p  cos  0  +  p  sin  0  =4;  (6) 
P  cos  0  —  3p  sin  0  =  6. 

_  6.  Write  in  polar  coordinates  x"^  +  2y^  —  4aj  =  0. 

71.  Identities  and  Conditional  Equations.     It  is  useful  to  make 
a  distinction  between  equalities  like 

(a  -  x){a  -\-  x)  =  a"^  -  x"^  (1) 

which  are  true  for  all  values  of  the  variable  x  and  equalities  like 

x^  -2x  =  Z  (2) 


§71]    THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS    133 

which  are  true  only  for  certain  particular  values  of  the  unknown 
number.  When  two  expressions  are  equal  for  all  values  of  the 
variable  for  which  the  expressions  are  defined,  the  equality  is 
known  as  an  identity.  When  two  expressions  are  equal  only  for 
certain  particular  values  of  the  unknown  number  the  equality  is 
spoken  of  as  a  conditional  equation.  The  fundamental 
formula 

sin^  0  +  cos^  0  =  1 
is  an  identity. 

2  sin  A  +  3  cos  A  =  3.55 
is  a  conditional  equation.     Sometimes  the  symbol  =  is  used  to 
distinguish  an  identity;  thus 

a^  —  x^  =  {a  —  x){a^  -\-  ax  -\-  x^) 

Exercises 

The  following  exercises  contain  problems  both  in  the  establishment 
of  trigonometric  identities  and  in  the  finding  of  the  values  of  the  un- 
known number  from  trigonometric  conditional  equations. 

The  truth  of  a  trigonometric  identity  is  established  by  reducing 
each  side  to  the  same  expression.  This  usually  requires  the  applica- 
tion of  some  of  the  fundamental  identities,  equations  (1)  to  (5), 
§53.  Facility  in  the  establishment  of  trigonometric  identities  is 
largely  a  matter  of  skill  in  recognizing  the  fundamental  forms  and  of 
ingenuity  in  performing  transformations.  In  verifying  the  identity 
of  two  trigonometic  expressions  it  is  best  to  reduce  each  exp  ression 
separately  to  its  simplest  form.  Unless  the  student  writes  the 
work  in  two  separate  columns,  transforming  the  left  member  alone 
in  one  column,  and  the  right  member  alone  in  the  other  column, 
he  is  very  liable  to  get  erroneous  results.  All  results  should  be 
checked.    The  following  worked  exercises  will  aid  the  student. 

(a)  Prove  that 

(1  —  sin  u  cos  M)(sin  u  +  cos  u)  =  sin^  u  +  cos'  u 
The  sum  of  two  cubes  is  divisible  by  the  sum  of  the  numbers  them- 
selves so  that  after  division  we  have: 

1  —  sin  u  cos  u  =  sin^  u  —  sin  u  cos  u  +  cos^  u 
Since  sin^  u  -\-  cos^  u  =  1,  this  equation  is  true  and  the  original  iden- 
tity is  established. 

(b)  Show  that 

sec^  X  —  1  =  sec^  x  sin^  x 


134        ELEMENTARY  MATHEMATICAL  ANALYSIS         [§71 

Substituting  sec^  x  = —  on  the  right  side 

cos^  X 

sin^  X 

sec^  X  —  \  = =  tan 2  x 

cos^  X 

or  sec2  X  =  l+tan^  x 

which  is  a  fundamental  identity. 

Solutions  to  exercises  in  trigonometric  conditional  equations  similar 

to  exercises  1,  4,  5,  9  below  must  be  checked.     The  necessity  for  a 

check  is  made  apparent  by  the  following  illustration : 

(c)  Solve  for  all  angles  less  than  360° 

2  sin  x  +  cos  x  =  2  (1) 

Transposing  and  squaring  we  get: 

oos^  a;=4  —  8sinx4-4  sin^  x  (2) 

since  sin^  x  +  cos^  x  =  \. 

1  —  sin^  aj  =  4  —  8  sin  a;  +  4  sin^  x  (3) 

5  sin2  a;  -  8  sin  re  +  3  =  0  (4) 

sin  X  =  1,  or  0.6  (5) 

X  =  90°,  37°,  or  143°  (6) 

Check:  2  sin  90°  +  cos  90°  =  2  +  0  =  2  (7) 

Check:  2  sin  37°  +  cos  37°  =  1.2  +  0.8  =  2  (8) 

Does  2  sin  143°  +  cos  143°  =  1.2  -  0.8  =  0.4  =2?  (9) 

The  last  value  does  not  check.  The  reasons  for  this  will  be  dis- 
cussed later  in  §§93  and  94.  Therefore  the  correct  solutions  are 
90°  and  37°. 

1.  Solve  for  all  values  of  ^   <  90°  :  6  cos^  ^  +  5  sin  0  =  7. 
Suggestion:    Write    6(1  —  sin^   0)  +  5  sin  6  =  7  and  solve   the 
quadratic  in  sin  6. 

6  sin2  0  -  5  sin  ^  +  1  =  0 
or, 

(3  sine  -  1  )(2sin  0  -  1)  =  0 
sin  0  =  1/3  or  1/2 
6  =  19°  or  30°. 

The  results  should  be  checked. 

2.  Prove  that  for  all  values  of  6  (except  7r/2  and  37r/2,  for  which 
the  expressions  are  not  defined) 

sec''  e  —  tan^  d  =  tan^  d  +  sec^  d. 


§71]    THE  CIRCLE  AND  THE  CIRCULAR  FUNCTIONS    135 

3.  Show  that 

sec^  u  —  sin^  u  =  tan'^  u  +  cos^  u, 
for  all  values  of  the  variable  u  except  90°  and  270°,  for  which  the 
expressions  are  not  defined.     , 

4.  Find  u,  if 

tan  u  +  cot  u  =  2. 

5.  Find  sec  ^,  if 

2  cos  0  +  sin  6=2. 

6.  Find  the  distance  of  the  end  of  the  diameter  of 

p  =  8  cos  (d  -  60°) 
from  the  line  OX. 

7.  If  pi  =  a  cos  6,  and  p2  =  a  sin  6,  find  pi   —  P2  when  d   =  60° 
and  a  =   5. 

8.  Find  the  polar  equation  of  the  circle  x^  -]-  y^  -{-  Qx  =  0. 

9.  For  what  value  of  6  does  p  =  3.55,  if  p  =  2  sin  5  +  3  cos  0? 
Result:     d  =  23°  30'  and  43°  30'. 

10.  Prove  that 

sin  A       _  1  +  cos  A 
1  —  cos  A  sin  A 

11.  Prove  that  2  cos^  u  —  1  =  cos'*  u  —  sin^  u. 

12.  Prove  that 

sec  u  +  tan  u 


sec  u  —  tan 

13.  Prove  that  sec^  u  +  csc^  u  =  csc^  u  sec^  u. 

14.  Show  that    (tan  a  +  cot  a)^  =  sec^  a  csc^  a. 

15.  Find  sin  d  if  esc  d  = 


16.  A  circular  arc  is  4.81  inches  long.  The  radius  is  12  inches. 
What  angle  is  subtended  by  the  arc  at  the  center?  Give  result  in 
radians  and  in  degrees. 

17.  Certain  lake  shore  lots  are  bounded  by  north  and  south  lines 
66  feet  apart.  How  many  feet  of  lake  shore  to  each  lot  if  the  shore- 
line is  straight  and  runs  77°  30'  E.  of  N.? 

18.  If  2/  =  2  sin  ^  +  3  cos  ^  -  3.55,  take  A  as  20°;  as  23°;  as  26°. 
Find  in  each  case  the  value  of  y.  From  the  values  of  y  just  found 
approximate  the  value  of  A  for  which  y  is  just  zero.  This  process  is 
known  as  "cut  and  try." 


136        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§71 

19.  The  line  y  =  (3/2) re  is  to  coincide  with  the  diameter  of  the 
circle : 

p  =  10  cos  {e  -  a) 
Find  a. 

20.  The  line  ?/  =  2x  is  to  coincide  with  the  diameter  of  the  circle : 

p  =  10  sin  {Q  +  a) 
Find  a. 

21.  To  measure  the  width  of  the  slide  dovetail  shown  in  Fig.  68, 
two  carefully  ground  cylindrical  gauges  of  standard  dimensions  are 
placed  in  the  F's  at  A  and  J5,  as  shown,  and  the  distance  X  carefully 


Fig.   68. — Diagram  to  Exercise  21. 

taken  with  a  micrometer.     The  angle  of  the  dovetail  is  60°.     Find 
the  reading  of  the  micrometer  when  the  piece  is  planed  to  the  required 
dimension  ilf  AT  =  4  inches.     Also  find   the   distance  F.     (Adapted 
from  "Machinery,"  N.  Y.) 
22.  Show  that: 

p  =  sin  0  +  cos  Q 


IS  a  circle. 

23    Draw  the 

curve : 

y  =  sin  X  +  cos  x. 

24    Sketch 

X 

y  =  2 

and 

y  =  Binx 

and  then 

?/  =  2  +  sin  a; 

and  discuss. 

CHAPTER  IV 
THE  ELLIPSE  AND  HYPERBOLA 


72.  The  Ellipse.  If  all  ordinates  of  a  circle  be  shortened  by 
the  same  fractional  amount  of  their  length,  the  resulting  curve 
is  called  an  ellipse.  For  example,  in  Fig.  69,  the  middle  points 
of  the  positive  and  negative  ordinates  of  the  circle  were  marked 
and  a  curve  drawn  through  the  points  so  selected.  The  result 
is  the  ellipse  A^A'B'^. 

If 

a;2  H-  2/2  =  a2  (1) 

is  the  equation  of  a  circle,  then 


x^  -\-  {my)'- 


(2) 


in  which  m  is  any  constant  >  1,  is  the  equation  of  an  ellipse;  for 
substituting  my  for  y  divides  all 
of  the  ordinates  by  m.  by  Theo- 
rem V  on  Loci,  §27.  The  ellipse 
may  also  be  looked  upon  as  the 
orthographic  projection  of  the 
circle.     See  §28. 

It  is  easy  to  show,  as  a  con- 
sequence of  the  above,  that  the 
shadow  cast  on  the  floor  by  a 
circular  disk  held  at  any  angle 
in  the  path  of  vertical  rays  of 
light  is  an  ellipse. 

The  curve  made  by  elongating 
by  the  same  fractional  amount 
of  their  length  all  of  the  abscissas  or  ordinates  of  a  circle  is  also 
an  ellipse,  as  the  following  considerations  will  show. 

First  let  the  ordinates  of  the  circle  (1)  be  shortened  as  before. 
The  result  is 

x2  -h  {myY  =  aP'  (2) 

137 


Fig.  69. — Definition  of  the  Ellipse. 


138        ELEMENTARY  MATHEMATICAL  ANALYSIS         [§72 

If  the  abscissas  of  the  same  given  circle  be  multiplied  by  rn 
to  make  another  curve,  the  result  is 


ii) 


+  y'  =  a'  (3) 


where  m  is  supposed  to  be  >  1  in  both  cases.  If  equation  (3) 
be  multiplied  through  by  m^  we  get: 

x^  +  (my)^  =  a'^m'^  (4) 

This  shows  that  the  second  curve  can  be  made  by  dividing  by  m 
all  of  the  ordinates  of  a  circle  of  radius  ma.  That  is,  (3)  is  an 
ellipse  made  from  a  circle  of  radius  ma  in  the  same  manner 
that  the  ellipse  (2)  is  made  from  a  circle  of  radius  a.  Hence  (3) 
is  an  ellipse  whose  dimensions  are  m-fold  those  of  (2). 

Thus  an  ellipse  results  if  all  of  the  ordinates  or  if  all  of  the  ab- 
scissas of  a  circle  be  multiplied  or  divided  by  any  given  constant  m. 
It  is  usual  to  write  the  multiplier  m  in  the  form  a/b,  so  that 
equation  (1)  may  be  written: 

x^  +  {ay /by  =  a2 
or: 

x2/a2  +  yVb2  =  1  (5) 

which  is  the  equation  of  the  ellipse  in  a  S3mimetrical  form.     Apply- 
ing the  principles  of  §27,  the  locus  (5)  may  be  thought  of  as 
made  from  the  unit  circle  x'^  +  2/^  =  1  by  multiplying  its  abscissas 
by  a  and  its  ordinates  by  b. 
When  written: 

y  =  ±  (b/a)  >l'a^-  x^  (6) 

2/  =   +    yja"^  -  x"^  (7) 

the  ellipse  and  circle  are  placed  in  a  form  most  useful  for  many 
purposes.  It  is  easy  to  see  that  (6)  states  that  its  ordinates  are 
the  fractional  amount  b/a  of  those  of  the  circle  (7). 

In  Fig.  69  the  points  A  and  A'  are  called  the  vertices  and  the 
point  0  is  called  the  center  of  the  ellipee.  The  line  ^A'  is  called 
the  major  axis  and  the  line  BB'  is  called  the  minor  axis.  It  is 
obvious  that  A  A'  =  2a,  and  from  equation  (5)  or  (6)  it  follows 
BB'  =  2b. 

The  definition  of  the  term  function  permits  us  to  speak  of  ?/  as  a 
function  of  x,  or  of  x  as  a  function  of  y,  in  cases  like  equation  (5) 


573] 


THE  ELLIPSE  AND  HYPERBOLA 


139 


above;  for  when  x  is  given,  ?/  is  determined.  To  distinguish  this 
from  the  case  in  which  the  equation  is  solved  for  y,  as  in  (6),  y,  in 
the  former  case,  is  said  to  be  an  implicit  function  of  x,  and  in  the 
latter  case  y  is  said  to  be  an  explicit  function  of  x. 

If  a  circular  cylinder  be  cut  by  a  plane,  the  section  of  the 
cyhnder  is  an  ellipse.  For  select  any  diameter  of  a  cir- 
cular section  of  the  cylinder  as  the  x-axis.  Let  a  plane  be  passed 
through  this  diameter  making  an  angle  a  with  the  circular  section. 
Then  if  ordinates  (or  chords  perpendicular  to  the  common  x-axis) 
be  drawn  in  each  of  the  two  planes,  all  ordinates  of  the  section 
made  by  the  cutting  plane  can 
be  made  from  the  ordinates  of 
the  circular  section  by  multiply- 
ing them  by  sec  a.  Hence  any 
plane  section  of  a  cylinder  is  an 
ellipse. 

73.  To  Draw  the  Ellipse.  A 
method  of  drawing  the  ellipse  is 
shown  in  Fig.  70.  Draw  con- 
centric circles  of  radii  a  and  b 
respectively,  a  >  b.  Draw  any 
number  of  radii  and  from  their 
intersections  with  the  larger 
circle  draw  vertical  lines,  and 
from  their  intersections  with  the  smaller  circles  draw  horizontal 
lines.  The  points  of  intersection  of  the  corresponding  horizontal 
and  vertical  lines  are  points  of  the  ellipse. 

Proof.     In  the  figure,  let  P  be  one  of  the  points  just  described. 
Then: 

P2D2 :  PiD  =  P2O  :  PiO 
or,  substituting  PD  for  the  equal  P2D2 

PD  :  P,D  =  P2O  :  PiO 
Now  OPi  =  a  and  OP2  =  b  and  PiD  is  the  ordinate  of  the  circle 
of  radius  a  or  is  equal  to  V  a^  —  x"^.     Substituting  these  in  the 
last  proportion  and  solving  for  PD  we  obtain: 

PD  =y  =  ±   -Va^  -  a:2 


-A  Construction  of  the 
Ellipse. 


140        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§74 

This  is  the  equation  of  an  ellipse.     Hence  the  curve  APB  is  an 

ellipse. 

-  The  two  circles  are  called  the  major  and  minor  auxiliary  circles. 

The  vectorial  angle  0  of  Pi  is  called  the  eccentric  angle  or  the 

eccentric  anomaly  of  the  point  P. 

Exercises 

1.  Draw  an  ellipse  whose  semi-axes  are  5  and  3,  and  write  its 
equation. 

2.  From  what  circle  can  the  ellipse  ?/  =  ±  ^\/9  —  x^  be  made  by 
shortening  of  its  ordinates? 

3  Write  the  equation  of  the  ellipse  whose  major  axis  is  7  and  minor 
axis  is  5. 

4.  Find  the  major  and  minor  axes  of  the  ellipse  x^/7  +  2/^/17  =  1. 

5.  What  curve  is  represented  by  the  equation  x^/9  +  y^/^6  =  1? 

74.  Parametric  Equations  of  the  Ellipse.  From  Fig.  70, 
OD  and  PD,  the  abscissa  and  ordinate  of  any  point  P  of  the 
ellipse,  may  be  written  as  follows: 

X  =  a  cos  ^  (1) 

y  =  b  sin  ^ 
for  OD  is  the  projection  of  OPi  =  a  through  the  angle  6  and  DP 
is  the  projection  of  OP 2  =  b  through  the  angle  ir /2  —  6.  The 
pair  of  equations  (1)  is  known  as  the  parametric  equations  of  the 
ellipse.  The  angle  0,  in  this  use,  is  called  the  parameter.  Writ- 
ing (1)  in  the  form: 

-  =  cos  0 
a 

I  =  sin  ^ 
0 

squaring,  and  adding,  we  eliminate  0  and  obtain: 

-  +  ^'  -  1  (2) 

the  symmetrical  equation  of  the  ellipse. 

If  the  abscissa  and  ordinate  of  any  point  of  a  curve  are  ex- 
pressed in  terms  of  a  third  variable,  the  pair  of  equations  are 
called  the  parametric  equations  of  the  curve.     Thus: 

X  =  U 


§74]  THE  ELLIPSE  AND  HYPERBOLA  141 

are  the  parametric  equations  of  a  certain  straight  line.  Its 
ordinary  equation  can  be  found  by  eliminating  the  parameter  t 
between  these  equations. 

From  equations  (1)  we  see  that  the  ellipse  might  be  defined  as 
follows:  Lay  off  distances  on  the  X-axis  proportional  tocos  6, 
and  distances  on  the  F-axis  proportional  to  sin  6.  Draw 
horizontal  and  vertical  lines  through  the  points  of  division,  thus 
dividing  the  rectangle  2a,  2b  into  a  large  number  of  small  rec- 
tangles. Starting  at  the  point  (a,  0)  and  drawing  the  diagonals 
of  successive  cornering  rectangles,  a  line  is  obtained  which  ap- 
proaches the  elli{)se  as  near  as  we  please  as  the  number  of  small 
rectangles  is  indefinitely  increased.  The  student  should  draw 
this  diagram.     See  Fig.  129. 

Exercises 

1.  Draw  the  curve  whose  parametric  equations  are: 

X  =  cos  d 
y  =  sin  e. 

2.  Write  the  equation  of  the  ellipse  whose  major  and  minor  axes 
are  10  and  6,  respectively. 

3  Find  the  axes  of  the  ellipse  whose  equation  is : 

4.  Write  the  parametric  equations  of  the  ellipse : 

y  =  ±  i  Vsi  -  xK 

6.  Discuss  the  curve : 

X  =   ±  h^4:  -  y\ 

6.  Discuss  the  curves: 

a;2  +  42/2  =  1 

4a:2  +    y2  =  1 

(1/4)^2  +    i/2  =  1. 

7.  Write  the  Cartesian  equation  of  the  curves  whose  parametric 
equations  are: 

,  .      fx  =  2  cos  5  ., .   Fx  =  6  cos  0  ,  .     [x  =  V3  cos  6 

^""^      ly  =     sin  d  ^^^  ly  =  2  sin  d  ^""^    ly  =  V2  sin  6. 

8.  What  locus  is  represented  by  the  parametric  equations 

X  =2t  +  l 
2/  =  3<  +  5? 


142        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§75 

9.  Show  that 

X  =  at 
y  =bt 
is  a  line  of  slope  b/a. 

10.  Write  the  equation  of  an  ellipse  whose  major  and  minor  axes 
are  6  and  4  respectively. 

11.  What  curve  is  represented  by  the  parametric  equations: 

X  =  2  +  Q  cos  9 
2/  =  5  +  2  sin  0? 

12.  Show  that  the  curve 

X  =  3  +  3  cos  0 
1/  =  2  +  2  sin  0 
is  tangent  to  the  coordinate  axes. 

13.  The  sunlight  enters  a  dark  room  through  a  circular  aperture  of 
radius  8  inches,  in  a  vertical  window  and  strikes  the  floor  at  an  angle 
of  60°.  Find  the  dimensions  and  the  equation  of  the  boundary  of 
the  spot  of  light  on  the  floor. 

14.  The  ellipse 

y  =  ±  i-v/9  -  x^ 
is  the  section  of  a  circular  cylinder.     Find  the  angle  a  made  by  the 
cutting  plane  and  the  axis  of  the  cylinder. 

75.1  Other  Methods  of  Constructing  an  Ellipse.  The  following 
methods  of  constructing  an  ellipse  of  semi-axes  a  and  b  may  be 
explained  by  the  student  from  the  brief  outlines  given: 

1.  Move  any  line  whose  length  is  a  -\-  b  (see  Fig.  71)  in  such  a 
manner  that  the  ends  A  and  B  always  lie  on  the  X-  and  F-axes, 
respectively.     The  point  P  describes  an  ellipse. 

2.  Mark  on  the  edge  of  a  straight  ruler  three  points  P,  M,  N, 
Fig.  72,  such  that  PM  =  b  and  PN  =  a.  Then  move  the  ruler 
keeping  M  and  N  always  on  A  A'  and  BB'  respectively.  P 
describes  an  ellipse.  The  elliptic  "trammel"  of  ''ellipsograph"  is 
constructed  on  this  principle  by  use  of  adjustable  pins  on  PMN  and 
grooves  on  A  A'  and  BB'. 

3.  Draw  a  semicircle  of  radius  a  about  the  center  C,  Fig.  73, 
and  produce  a  radius  to  0  such  that  CTO  =  a  -\-  b.     From  C  draw 

iThis  section  may  be  omitted  altogether  or  assigned  as  problems  to  various 
members  of  the  class. 


§75] 


THE  ELLIPSE  AND  HYPERBOLA 


143 


any  number  of  lines  to  the  tangent  to  the  circle  at  T.  From  0 
draw  lines  meeting  the  tangent  at  the  same  points  of  TN.  At  the 
points  where  the  lines  from  C  cut  the  semicircle,  draw  parallels 


-^ 

B 

/ 

\m 

O 

\a 

V 

-^ 

^ 

^"  ^ 

J 

B' 

Fig.  71.— Ellipse     Traced    by 
a  Point  P  of  the  Moving  Line  AB. 


Fig.  72. — Theory  of  the  Common 
Ellipsograph"  or  Elliptic  Trammel. 


Fig.  73. — A  Graphical  Construction  of  an  Ellipse. 

to  CT.    The  points  of  meeting  of  the  latter  with  the  lines  radiating 
from  0  determine  points  on  the  ellipse. 


144        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§76 

To  prove  the  above,  note  that  OB  =  a  cos  6,  PD  =  OD  tan  d', 
also  that  tan  d  :  tan  6' :  :a  :  b.  Discuss  the  latter  case  when  b  =  a 
and  also  when  b  >  a. 

76.  Origin  at  a  Vertex.  The  equations  of  the  ellipse  (5)  and  (6) 
§72  and  (1)  §74  are  the  most  useful  forms.  It  is  obvious 
that  the  ellipse  may  be  translated  to  any  position  in  the  plane 
by  the  methods  already  explained.  The  ellipse  with  center 
moved  to  the  point  {h,  k)  has  the  equation: 

{X  -  hy    (y-ky_ 

a2       "^       62       ~  ■"  ^^^ 

Of  special  importance  is  the  equation  of  the  ellipse  when  the  origin 
is  taken  at  the  left-hand  vertex.  This  form  is  best  obtained  from 
equation  (6),  §72,  by  translating  the  curve  the  distance  a  in 
the  X  direction.     Thus: 


or, 


or,  letting  I  stand  for  the  coefficient  of  x, 

y^  =  lx-  -'a:^  =  lx{l  -  x/2a)  (2) 

u 

For  small  values  of  x,  x  /2a  is  very  small  and  the  ellipse  nearly 
coincides  with  the  parabola  y^  =  Ix. 

77.  Any  equation  of  the  second  degree,  lacking  the  term  xy  and 
having  the    terms    containing  x^  and  y"^  both    present   and  with 
coefficients  of  like  signs,  represents  an  ellipse  with  axes  parallel  to 
the  coordinate  axes.     This  is  readily  shown  by  putting  the  equation 
ax^  +  by^  +  2gx  +  2fy+  c  =  0  (1) 

in  the  form  (1)   of  the  preceding  section.     The  procedure  is  as 
follows: 

aix''  +  2^x)  +  6(2/2  +  2-^2/)  =   -c  (2) 


=  .+ 

i^^- 

y 

-{X- 

ar 

2/2: 

2b^ 

§78]  THE  ELLIPSE  AND  HYPERBOLA  145 

Let  M  stand  for  the  expression  in  the  right-hand  member  of  (3), 
then  we  get: 

M  M 

a  1 

This  shows  that  (1)  is  an  eUipse  whose  center  is  at  the  point 

( }  —  X I  and  which  is  constructed  from  the  circles  whose  cen- 

a  0/ 

ters  are  at  the  same  point  and  whose  radii  are  the  square  roots  of  the 
denominators  in  (4).  The  major  axis  is  parallel  to  OX  or  OF  ac- 
cording as  a  is  less  or  greater  than  b.  The  cases  when  the  locus 
is  not  real  should  be  noted.     Compare  §42. 

Illustration:     Find  the   center  and   axes   of  the  elllipse 

x^  +  4i/2  -{-Qx  -  8y  =  23 
Write  the  equation  in  the  form 

a;2  +  6x  +  4?/2  -  Sy  =  23 
Complete  the  squares 

x2  +  6x  +  9  +  42/2  -  8?/  +  4  =  36 
Rewriting  (a;  +  3)^  +  4(2/  -  1)2  =  36 

or  (a; +  3)2/36  +  {y  -  1)2/9  =  1 

This  is  seen  to  be  an  ellipse  whose  center  is  at  the  point  (—  3,  1) 
and  whose  semi-axes  are  a  =  6  and  6  =  3. 

The  rotation  of  the  ellipse  through  any  angle  about  0  as  a 
center  will  be  considered  in  another  place.  It  should  be  noted, 
however,  that  the  ellipse  is  turned  through  90°  by  merely  inter- 
changing X  and  y. 

78.  Limiting  Lines  of  an  Ellipse.  It  is  obvious  from  the 
equation 

2/  =  ±  -  Va2  -  x2 

that  X  =  a  and  x  =  —  a  are  hmiting  hues  beyond  which  the  curve 
cannot  extend;  that  is,  x  cannot  exceed  a  in  numerical  value 
without  y  becoming  imaginary.  The  same  test  may  be  apphed 
to  equations  of  the  form : 

a:2  +  4a:  +  9?/2  -  O?/  +  4  =  0 

10 


146        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§78 

Solving  for  y  in  terms  of  x: 

Sy  =  1  ±   Vr-  (a;  +  2)2 
The  values  of  y  become  imaginary  when: 
(x  +  2)2  >  1 
or, 

a:  +  2>4-lor<-l 
or, 

x>-lor<-3 

These,  then,  are  the  limiting  lines  in  the  x  direction.  Finding 
the  limiting  lines  in  the  y  direction  in  the  same  way,  the  rectangle 
within  which  the  ellipse  must  lie  is  determined. 

In  cases  like  the  above  the  actual  process  of  finding  the  limiting 
lines  and  the  location  of  the  center  of  the  ellipse  is  best  carried 
out  by  the  method  of  §77. 

Exercises 

Find  the  lengths  of  the  semi-axes  and  the  coordinates  of  the  center 
for  the  six  following  loci  and  translate  the  curves  so  that  the  terms  in 
X  and  y  disappear,  by  the  method  of  §77. 

1.  12x2  -  48x  +  32/2  +Qy  =  13. 

2.  2/'  -  8?/  +  4^2  +  6  =  0. 

3.  x2  -  6x  +  4?/2  +  Sy  =  5.        , 

4.  x2  +  9t/2  -  12x  +  Qy  =  12. 

6.  4x2  +  2/2  -  12x  +  2i/  -  2  =  0. 

6.  x2  +  2i/2  -  X  -  V2y  =  1/2. 

7.  Show  that  x2  +  4x  +  9y^  —  Qy  =  0  passes  through  the  origin. 

8.  Show  that  x^  —  4x  +  4y^  -\-  Sy  +  4:  =  0  is  an  ellipse. 

9.  Discuss  the  curves : 

x'      y'  _ 
9  "^4 

9  "^  4 
^2      y^  _ 
4  "^  9 

2 


+  ^  =  0. 


9    '   4 


§79]  THE  ELLIPSE  AND  HYPERBOLA  147 

10.  Discuss  the  following  parabolas : 

y  =  2'px^ 
y  =  —  2px2 

y  =  j^^x^ 

y  =  -  2px^  +  b. 
What  are  the  roots  of  the  last  function? 

11.  Write  the  symmetrical  equation  of  the  ellipse  if  its  parametric 
equations  are : 

X  =  (3/2)  cos  d 
y  =  (2/3)  sin  d. 

12.  Discuss  the  curve  y^  =  (18/5)x  -  (9/25) x^. 

13.  Compare  the  curves  ij"^  =  x  —  x^  and  y"^  =  x. 

14.  Find  the  center  of  the  curve  y"^  =  2x  {Q  —  x). 

79.  Graph  of  y  =  tan  x.  If  this  graph  is  to  be  constructed  on  a 
sheet  of  ordinary  letter  paper,  8§  inches  X  1 1  inches,  it  is  desirable 
to  proceed  as  follows :  Draw  at  the  left  of  the  sheet  of  paper  a  semi- 
circle of  radius  L15  .  .  .  inches,  (that  is,  of  radius  =  IS/Stt),  so 
that  the  length  of  the  arc  of  an  angle  of  10°  or  tt  /18  radians  will  be 
1  /5  inch.  Take  for  the  x-axis  a  radius  COX  prolonged,  and  take  for 
the  y-Sixis  the  tangent  OY  drawn  through  0,  as  in  Fig.  74.  Divide 
the  semicircle  into  eighteen  equal  parts  and  draw  radii  through  the 
points  of  division  and  prolong  them  to  meet  OF  in  points  Ti,  T^, 
Tz,  Ti,  .  .  .  Then  on  the  y-axis  there  is  laid  off  a  scale 
YY'  in  which  the  distances  OTi,  OT2,  .  .  .  are  proportional  to 
the  tangents  of  the  angles  OCSi,  OCS2,  .  .  . ;  f or  the  tangents 
of  these  angles  are  OTi/CO,  OT2/CO,  ...  and  CO  is  the  unit  of 
measure  made  use  of  throughout  this  diagram.  Draw  horizontal 
lines  through  the  points  of  division  on  07  and  vertical  lines  through 
the  points  of  division  on  OX,  thus  dividing  the  plane  into  a  large 
number  of  small  rectangles.  Starting  at  0,  tt,  2x,  .  .  .  —  tt, 
—  27r,  .  .  .  and  sketching  the  diagonals  of  consecutive  cornering 
rectangles,  the  curve  of  tangents  is  approximated.  Greater  pre- 
cision may  be  obtained  by  increasing  as  desired  the  number  of 
divisions  of  the  circle  and  the  number  of  corresponding  vertical 
and  horizontal  lines. 

It  is  observed  that  the  graph  of  the  tangent  is  a  series  of  similar 
branches,  which  are  discontinuous  for  x  =  tt  12,  —ir  12,  (3/2)7r, 


148        ELEMENTARY  MATHEMATICAL  ANALYSIS 


580 


—  (3  /2)7r,  .  .  .  For  these  values  of  x  the  curve  has  vertical 
asymptotes,  as  shown  at  AB,  A'B',  in  Fig.  74. 

If  the  number  of  corresponding  vertical  and  horizontal  lines 
be  increased  sufficiently,  the  slope  of  the  diagonal  of  any  rectangle 
gives  a  close  approximation  to  the  true  slope  of  the  curve  at  that 
point. 

It  has  already  been  noted  that  all  of  the  trigonometric  functions 
are  periodic  functions  of  period  27r.     It  is  seen  in  this  case,  however, 


Y                     M    A                                                      M'  JL 

/        '       L                            '-   1 

//i  \  I                              \  I 

Q 

rM/^     I                                  I 

J/3c''^  ^         ::::::":z:^::::::±:::: 

i///\    IS                   z     "^ 

//Z>>^W.            \                           z 

Wi^ — /                 ^                      7                  \                 

i|fe^-^~o                 -'f\                  /^"T                Jf^                  Z. 

^5^^                 -    ^             Z                   T^            Z 

\V^                           i        Z                      "        \      "Z" 

\vX^      -          -^-^                          -^-^ 

^\\                               t                                       h       - 

\\    I      V\             i\ 

\     w     \  \ 

B     N' 


B     N' 


27r- 


Fig.  74. — Graphical  Construction  of  the  Curve  of  Tangents  y  =  tana;. 
For  lack  of  room  only  a  few  of  the  points  Si,  S2,  ...  Ti,  T2,,  .  .  ■  are  lettered 
in  the  diagram.     The  dotted  curve  is  y  =  cot  x. 


that  tan  x  has  also  the  shorter  period  w ;  for  the  pattern  MN, 
M'N',  M"N'\  of  Fig.  74  is  repeated  for  each  interval  tt  of  the 
variable  x. 

80.  Ratio  (sin  x)/x  and  (tan  x)/x  for  Small  Values  of  x.  Pre- 
cisely as  in  the  case  of  the  locus  of  y  =  sin  x,  the  rectangles  along, 
and  on  both  sides  of,  the  x-axis  in  the  graph  of  2/  =  tan  x,  are 
nearly  squares.     In  Fig.  74,  the  a^-sides  of  these  rectangles  are 


§80]  THE  ELLIPSE  AND  HYPERBOLA  149 

1  /5  inch,  but  the  ?/-sides  are  sHghtly  greater,  since  OTi  is  slightly 
greater  than  the  arc  OSi  of  the  circle.  To  prove  this,  note  that  OTi 
is  half  of  one  side  of  a  regular  18-sided  polygon  circumscribed  about 
the  circle;  since  the  perimeter  of  this  polygon  is  greater  than  the 
circumference  of  the  circle,  OTi  >  OSi,  for  these  magnitudes  are 
1  /36  of  the  perimeter  and  circumference,  respectively,  just  named. 
Likewise  in  Fig.  59,  DSi  <  OSi,  for  DSi  is  one-half  of  the  side  of 
an  18-sided  regular  polygon  inscribed  in  the  circle  and  OSi  is 
1  /36  of  the  circumscribed  circumference. 
Hence: 

sin  a;  <  a;  <  tan  x  (1) 

or  dividing  by  sin  x, 

1  <  -A-  <  sec  X  (2) 

sm  x  ^  ' 

Now  as  X  approaches  0,  the  last  term  of  this  inequahty  approaches 
unity.  Hence  the  second  term,  whose  value  always  lies  between 
the  first  and  third  term  of  the  inequahty,  must  approach  the  same 
value,  1.     This  fact  is  expressed  in  mathematics  by  the  statement 

the  limit  of  =  1  as  a:  approaches  0 

X 

or,  in  symbols: 

lim       sin  x 
X  =  0 

Dividing  (1)  by  tan  x, 


^     =1  (3) 


X 

cosx  <  7 — —  <  1  (4) 

tan  X  ^  ' 

Now  as  X  approaches  0,  the  first  term  of  this  inequahty  approaches 
unity.  Hence  the  second  term,  whose  value  always  lies  between 
the  first  and  third  term  of  the  inequality,  must  approach  the  same 
value,  1.     This  fact  is  expressed  by  the  statement 

the  limit  of =  1  as  .t  approaches  0 

or,  in  symbols: 

lim       tanx  ^ 
X  =  0        X  ^  ^^^ 

Equations  (3)  and  (5)  express  very  useful  and  important  facts. 
Geometrically  they  state  that  the  rectangles  along  the  a;-axis  in 


150        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§81 

Figs.  59  and  74,  approach  more  and  more  nearly  squares  as  the 
number  of  intervals  in  the  circle  is  increased.  Each  of  the  ratios 
in  (2)  approaches  as  near  as  we  please  to  unity  the  smaller  x  is 
taken,  but  the  limits  of  these  ratios  are  unity  only  when  the  angles, 
are  measured  in  radians. 

The  word  "limit"  used  above  stands  for  the  same  concept  that 
arises  in  elementary  geometry.  It  may  be  formally  defined  as 
follows : 

Definition:  A  constant,  a,  is  called  the  limit  of  a  variable, 
t,  if,  as  t  runs  through  a  sequence  of  numbers,  the  difference 
(a  —  0  becomes,  and  remains,  numerically  smaller  than  any  pre- 
assigned  number. 

81.  Graph  of  cot  x.  In  order  to  lay  off  a  sequence  of  values  of 
cot  ^  on  a  scale,  it  is  convenient  to  keep  the  denominator  con- 
stant in  the  ratio  (abscissa)  /(ordinate)  which  defines  the  cotangent. 

Pn  Pio     ^9  ^8  P',  P^  P,     Pz        P2  Pi 


^ 

3 

^ 

1 

1 

1 

ji 

f 

^ 

^ 

r>ii  Z>io     DjD^D',         O         D{,Di    Ds        D2  D\ 

Fig.  75. — Construction  of  a  Scale  of  Cotangents. 

The  denominator  may  also,  for  convenience,  be  taken  equal  to 
unity.  Thus,  in  Fig.  75,  the  triangles  of  reference  DiOPi,  D2OP0, 
.  .  .for  the  various  values  of  9  shown,  have  been  drawn  so  that 
the  ordinates  PiDi,  P2D2,  ...  are  equal.  If  the  constant  ordi- 
nate be  also  the  unit  of  measure,  then  the  sequence  QDi,  OD2,  OD^, 
.  .  .  OD7,  ODs,  represents,  in  magnitude  and  sign,  the  cotan- 
gents of  the  various  values  of  the  argument  6.  Using  ODi,  OD2, 
...  as  the  successive  ordinates  and  the  circular  measure  of 
0  as  the  successive  abscissas,  the  graph  oi  y  =  cot  .t  is  drawn,  as 
shown  by  the  dotted  curve  in  Fig.  74. 

The  sequence  ODi,  OD2,  .  .  .  Fig.  75  is  exactly  the  same  as  the 
sequence  OTi,  OT2,  .  .  .  Fig.  74,  but  arranged  in  the  reverse 
order.  Hence,  the  graph  of  the  cotangent  and  of  the  tangent  are 
alike  in  general  form,  but  one  curve  descends  as  the  other  ascends, 
so  that  the  position,  in  the  plane  xy,  of  the  branches  of  the  curve 


m] 


THE  ELLIPSE  AND  HYPERBOLA 


151 


are  quite  different.  In  fact,  if  the  curve  of  the  tangents  be  rotated 
about  OF  as  axis  and  then  translated  to  the  right  the  distance 
TT  /2,  the  curves  would  become  identical.  Therefore,  for  all  values 
of  x: 

tan  (7r/2  —  x)  =  cot  a;  (1) 

This  is  a  result  previously  known. 

Y  N    A  N" 


1 "" 

\\\      y            ■        \ 

\\\\         /p                                       \ 

>^XXM-     ^^                  -                                       -^^   - 

''^^S^Mo 

iw^^^r—-                                     '^                                        TT                                    37r 
V^V^                                 2                                                                       IT 

yfti-\ 7^ "s— ^ 

///                    /           \ 

Z7T__:\    "T"       "^    _.„___.. 

n__..\\\ 

Y'  B    N'  N' 

Fig.  76. — Graphical  Construction  oi  y  =  sec  x. 

82.  Graph  of  y  =  sec  x.  Since  sec  6  is  the  ratio  of  the  radius 
divided  by  the  abscissa  of  any  point  on  the  terminal  side  of  the 
angle  6,  it  is  desirable,  in  laying  off  a  scale  of  a  sequence  of  values 
of  sec  6,  to  draw  a  series  of  triangles  of  reference  with  the  abscissas 
in  all  cases  the  same,  as  shown  in  Fig.  76.  In  this  figure  the  angles 
were  laid  off  from  CQ  as  initial  line.     Thus: 

CT./CS,  =  secQC*S5 

or,  if  CSi  be  unity,  the  distances  like  CTs,  laid  off  on  CQ,  are  the 


152        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§83 

secants  of  the  angles  laid  off  on  the  arc  QS^O  or  laid  off  on  the  axis 
OX. 

The  student  may  describe  the  manner  in  which  the  rectangles 
made  by  drawing  horizontal  lines  through  the  points  of  division  on 
CQ  and  the  vertical  lines  drawn  at  equal  intervals  along  OX,  may 
be  used  to  construct  the  curve.  If  the  radius  of  the  circle  be  1.15 
inches,  what  should  be  the  length  of  Ow  in  inches? 

The  student  may  construct  and  discuss  the  locus  oi  y  =  esc  x. 
Compare  with  the  locus 

2/  =  sec  a; 

Exercises 

1.  Discuss  from  the  diagrams,  59,  74,  76,  the  following  statements: 
Any  number,  however  large  or  small,  is  the  tangent  of  some  angle . 
The  sine  or  cosine  of  any  angle  cannot  exceed  1  in  numerical  value. 
The  secant  or  cosecant  of  any  angle  is  always  numerically  greater 

than  1  (or  at  least  equal  to  1). 

2.  Show  that  sec  (w  —  x]    =  esc  .t  for  all  values  of  x. 


(I-) 


by  use  of  polar  coordinate  paper.  Form  M3. 

4.  Describe  fully  the  following,   locating  nodes,  troughs,  crests, 
asymptotes,  etc.: 

y  =  sin   \x  -  ^ j 
y  =  cos  Ix  +  ^j 

y  =  tan  ^x  +  ^j 

y  =  tan  (a;  +  1). 
83.  Increasing  and  Decreasing  Functions.  The  meanings  of 
these  terms  have  been  explained  in  §26.  Applying  these  terms  to 
the  circular  functions,  we  may  say  that  y  =  smx,y  =  tana:, 
y  =  secx  are  increasing  functions  for  0  <  a;  <  tt  /2.  The  co- 
functions,  y  =  cos  x,y  =  cot  x,  y  =  esc  x,  are  decreasing  functions 
within  the  same  interval. 

Exercises 

Discuss  the  following  topics  from  a  consideration  of  the  graphs  of 
the  functions : 


}84] 


THE  ELLIPSE  AND  HYPERBOLA 


153 


1.  In  which  quadrants  is  the  sine  an  increasing  function  of  the 
angle?     In  which  a  decreasing  function? 

2.  In  which  quadrants  is  the  tangent  an  increasing,  and  in  which  a 
decreasing,  function  of  its  variable? 

3.  In  which  quadrants  are  the  cos  d,  cot  d,  sec  6,  esc  6,  increasing 
and  in  which  are  they  decreasing  functions  of  0? 

4.  Show  that  all  the  co-functions  of  angles  of  the  first  quadrant  are 
decreasing  functions. 

J 


Construction  of  the  Rectangular  Hyperbola. 


84.  The  Rectangular  H3rperbola.  We  have  seen  that  the  circle 
is  the  locus  of  a  point  whose  abscissa  is  a  cos  d  and  whose  ordinate 
is  a  sin  6.  The  rectangular,  or  equilateral,  hyperbola  may  be 
defined  to  be  the  locus  of  a  point  whose  abscissa  is  a  sec  6  and  whose 
ordinate  is  a  tan  6.  To  construct  the  curve,  divide  the  X-axis  pro- 
portionally to  sec  6,  and  the  F-axis  proportionally  to  tan  0,  as 
shown  in  Fig.  77.  The  scale  OX  of  this  diagram  may  be  taken 
from  OF  of  Fig.  76,  and  the  scale  OY  may  be  taken  from  OF  of  Fig. 


154        ELEMENTARY  MATHE^MATICAL  ANALYSIS        [§8G 

74.     The  plane  of  xy  may  be  divided  into  a  large  number  of  rec- 
tangles by  passing  lines  through  the  points  of  division  perpendicu- 
lar to  the  scales  and  then,  starting  from  A  and  A',  sketching  the 
diagonals  of  the  successive  cornering  rectangles. 
•The  parametric  equations  of  the  curve  are,  by  definition: 

X  =  a  sec  ^1 

y  =  a  tan  e\  ^^^ 

The  Cartesian  equation  is  easily  found  by  squaring  each  of  the 
equations  and  subtracting  the  second  from  the  first,  thus  eliminat- 
ing d  by  the  relation  sec^  B  —  tan^  ^  =  1 : 

a;2  _  y2  =  a2(sec2  e  -  tan2  6) 
or, 

x2  -  y2  =  a2  (2) 

This  is  the  Cartesian  equation  of  the  rectangular  hyperbola. 

The  equation  of  the  rectangular  hyperbola  may  also  be  written  in 
the  useful  form : 

y  =  +  Vx2  -  a2  (3) 

Compare  (1)  and  (3)  with  the  equations  of  the  circle. 

The  rectangular  hyperbola  here  defined  will  be  shown,  in  §86, 
to  be  the  curve  2xy  =  a^  rotated  45°  clockwise  about  the  origin. 

85.  The  Asymptotes.  Let  G'G  be  the  Hne  y  =  x,  Fig.  77.  The 
slope  of  OP^  is  PD  jOD  or  y  jx  ov 

a  tan  d 

a^ol  =  ^'"  "• 
The  value  of  6  corresponding  to  the  point  P  is  AOH.  As  the  point 
P  moves  upward  and  to  the  right  on  the  curve,  the  angle  6,  or 
AOH,  approaches  90°  and  sin  B  approaches  unity.  Hence  the 
line  OP  approaches  OG  as  a  limit,  and  P  approaches  as  near  as  we 
please  to  OG.  The  same  reasoning  applies  to  points  moving  out 
on  the  curve  in  the  other  quadrants.  The  lines  GG'  and  J  J'  are 
called  asymptotes  to  the  hyperbola. 

86.  The  Curves  2xy  =  a^  and  x^  -  y2  =  a^.  In  Fig.  78,  let 
the  curve  be  the  locus  2xiyi  —  a^,  referred  to  the  axes  X/Xi  and 
y/Fi.  This  curve  has  already  been  called  the  rectangular  hyper- 
bola.    (See  §23.)     We  desire  to  find  the  equation  of  the  curve 

1  To  avoid  an  excessive  number  of  construction  lines,  OP  is  not  shown  in  the 
figure. 


m 


THE  ELLIPSE  AND  HYPERBOLA 


155 


In  the  figure,  ?/i  is  the  sum 
The  angle  of  projection  is 


(1) 

°   of  X2 


referred  to  the  axes  X2X'2  and  Y2Y\ 
of  the  projections  of  X2  and  2/2  on  PDi. 
45°,  whose  cosine  is  2\/2.    Hence, 

Vi  =  2^2{y2  +  X2) 
Likewise,  Xy  is  the  difference  in  the  projections  through  45' 
and  2/2  on  XiX'i.     Or: 

^1  =  h\/2{x2  -  Vi) 
Hence,     multiplying     the 
members  of  (1)  and  (2): 

2xiyi  =  ^2^  -  2/2^  (3) 
Since  by  hypothesis  2xiyi 
=  a^,  the  equation  of  the 
curve  referred  to  the  axes 
X272  is 

X2''  -  2/2^  =  a^       (4) 
Thus,    2xy    =    a^    is    the 


2xy  = 


anti-clockwise  through  an 
angle  of  45°. 

By  §27,  the  curve  2xy 
=  a^  may  be  made  from 
xy  =  1  by  multiplying 
both  the  abscissas  and  the  ordinates  by  a/\/2. 

Are  the  curves  xy  =  1  and  x^  —  y^  =  1  oi  the  same  size  ? 

87.  Hyperbola  of  Semi-axes  a  and  b.  The  curve  whose  ab- 
scissas are  proportional  to  sec  0  and  whose  ordinates  are  pro- 
portional to  tan  0  is  called  the  hj^erbola.  Its  parametric 
equations  are,  therefore: 

X  =  a  sec  ^1 

y  =  b  tan  ^J  ^^^ 

where  a  and  b  are  constants. 

To  construct  the  curve,  draw  two  concentric  circles  of  radii  a  and 
b,  respectively,  as  in  Fig.  79.  Divide  both  circumferences  into 
the  same  number  of  convenient  intervals.  Lay  off,  on  XOX', 
distances  equal  to  a  sec  0  by  drawing  tangents  at  the  points  of 
division  on  the  circumference  of  the  a-circle;  also  lay  off  distances 


156        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§87 

equal  to  b  tan  6  on  the  vertical  tangent  to  the  6-circle  by  prolong- 
ing the  radii  of  the  latter  through  the  points  of  division  of  the  cir- 
cumference. Draw  horizontal  and  vertical  lines  through  the 
points  of  division  of  MN  and  XX^  respectively,  dividing  the 
plane  into  a  large  number  of  rectangles  which  are  used  exactly 
as  in  Fig.  77  for  the  construction  of  the  curve. 

In  the  above  construction,  there  is  no  reason  why  the  diameter  of 
the  6-circle  may  not  exceed  that  of  the  a-circle. 


J 

s 

V 

N 

T         / 

N 

y 

5 

^ 

\. 

T 

y 

V 

p 

\ 

Y 

K 

/ 

\ 

/ 

/ 

I 

M 

u 

A 

D 

/ 

I 

1 

\ 

/ 

x 

< 

\ 

^ 

/' 

\ 

\ 

G 

\> 

V, 

> 

-X 


Fig.  79.— The  Hyperbola  x'^/a'^  -  y^/b^  =  1. 
Writing  (1)  in  the  form: 


=  sec  6 
=  tan  d 


and  eliminating  6  as  before  we  obtain: 
x2      y" 


b2 


=  1 


(2) 


§87]  THE  ELLIPSE  AND  HYPERBOLA  157 

the  Cartesian  equation  of  the  hyperbola.     This  is  also  called  the 
symmetrical  equation  of  the  hyperbola. 

The  line  A  A'  =  2a  is  called  the  transverse  axis,  the  line  BB' 
is  called  the  conjugate  axis,  the  points  A  and  A'  are  called  the 
vertices,  and  the  point  0  is  called  the  center  of  the  hyperbola. 
Let  the  line  G'OG  be  the  line  through  the  origin  of  slope  b  fa  and  let 
J'OJ  be  the  line  of  slope  —  b  fa.  The  slope  of  the  radius  vector 
OP  is: 

PD       y        bi&nd       b    .    ^ 

7^^   =    —   = ;:   =    —  Sin  ^ 

OD  X  asec  d  a 
The  limit  of  this  ratio  as  the  point  P  moves  out  on  the  curve  away 
from  Oisb/a;  for  d  approaches  90°  as  P  moves  outward,  and  hence 
sin  6  approaches  1.  Hence,  the  line  OP  approaches  in  direction 
OG  as  a  limit.  Points  moving  along  the  curve  away  from  0  in  the 
other  quadrants  likewise  approach  as  near  as  we  please  to  G'G  or 
J' J.  The  lines  G'G  and  J' J  are  called  the  asymptotes  of  the  hyper- 
bola.    The  equations  of  these  lines  are 

y=  +  ^x  (3) 

Solving  the  equation  (2)  for  y,  the  equation  of  the  hyperbola  may 
be  written  in  the  useful  form 

y  =  ±  ^Vx'-^'  (4) 

Compare  this  equation  with  the  equation  of  the  ellipse,  (6)  §  72. 

It  is  easy  to  show  that  the  vertical  distance  PG  of  any  point  of 
the  curve  from  the  asymptote  G'G  can  be  made  as  small  as  we  please 
by  moving  P  outward  on  the  curve  away  from  0. 

Write  the  equation  of  the  hyperbola  in  the  form 

yi  =  -\/^"2"-~a^  ,  (5) 

and  the  equation  of  the  asymptote  GG'  in  the  form 

2/2  =  Ix  (6) 

Then: 

PG  =  y2-  yx  =  ^  (x  -  Vx^  -  a^)  (7) 


a 


a  X  -\-  \/x^  —  a'' 


(8) 


158        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§88 

by  multiplying  both    numerator    and    denominator  in   (7)   by 
X  +  \/x^  —  o^.      Now,  as  X  increases  in  value  without  limit  the 
right  side  of  (8)  approaches  zero.     Whence: 
P(?  =  0  as  a:  =  00 

Exercises 

1.  Write  the  symmetrical  equation  of  the  hyperbola  from  the 
parametric  equations  x  =  5  sec  6,  y  =  3  tan  d. 

2.  Find  the  Cartesian  equation  of  the  hyperbola  from  the  relations 
X  =  7  sec  6,  y  =10  tan  6.  Note  that  the  graphical  construction  of 
the  hyperbola  holds  if  6  >  a. 

3.  What  curve  is  represented  by  the  equation 

(X  -  Sy  _  (y  +  2y  _    ^ 
25  16        ~  ^• 

4.  What  curve  is  represented  by  the  equation  y  =  ^Vx"^  —   a^? 

5.  Write  the  equation  of  a  hyperbola  having  the  asymptotes 
y  =  ±  (3/4)  X,  and  transverse  axis  =  24. 

6.  Show  that  the  curves 

x^  -\-  Qx  -  y^  -  4:y  +  4:  =  0 
and 

{x  +  3)2  -  (^  +  2)2  =  1 

are  the  same,  and  show  that  each  is  a  hyperbola. 

7.  What  curve  is  represented  by  the  equations 

X  =  h  -\-  a  sec  d 
2/  =  fc  +  &tan  0? 

8.  Discuss  the  curve  x^  —  Sx  —  2y^  —  12y  =  0. 

88.  Orthographic  Projections.  When  the  equation  of  the 
hyperbola  is  written  in  the  useful  form 

y  =  ±  ^Vx^^^^  (1) 

it  is  seen  that  the  hyperbola  may  be  looked  upon  as  generated  from 
the  equilateral  hyperbola 

y  =  ±  \/x^  -  a^  (2) 

by  multiplying  all  of  its  ordinates  hyb/a. 

89.  Conjugate   Hyperbolas.     Consider   the   hyperbola 

^'-^  =  1  m 


§89] 


THE  ELLIPSE  AND  HYPERBOLA 


159 


Interchanging  x  and  y  in  this  equation  gives,  by  Theorem  III  on 
Loci,  §24,  a  new  locus  which  is  the  reflection  of  (1)  in  the  line 
y  =  X.     The  new  equation  may  be  written  in  the  form 


y 


1 


(2) 


in  which  all  signs  have  been  changed  after  interchanging  x  and  y. 
Since  (2)  is  the  same  curve  as  (1)  but  in  a  new  position,  it  is  still 
a  hyperbola;  its  vertices  are  located  on  the  F-axis  instead  of  on 
the  X-axis.     The  asymptotes  of  (1)  have  been  found  to  be 


y  = 


Therefore  the  asymptotes  of 

(2)  may  be  found  by  reflecting 

(3)  in  the  line  y  =  x;  hence 
they  must  be  given  by: 


±^x 


(4) 


Now,  if  the  constants  a 
and  b  in  equation  (2)  be  in- 
terchanged giving  thereby  the 
equ  ation 

then  the  shape  of  the  hyper- 
bola (2)  will  be  changed  but 
its  position  will  be  unaltered, 
that  is,  its  vertices  will  still 
be  located  on  the  F-axis. 
The  asymptotes  of  (5)  are 
found,  of  course,  by  inter- 
changing a  and  h  in  (4),  which 

gives  an  equation  exactly  like  (3).  Hence  the  hyperbola  (5)  has 
the  same  asymptotes  as  the  original  hyperbola  (1).  When  a 
hyperbola  with  vertices  on  the  F-axis  has  the  same  asymptotes 
as  a  hyperbola  with  vertices  on  the  X-axis,  and  of  such  size  that 
the  transverse  axis  of  one  hyperbola  is  the  conjugate  axis  of  the 
other,  then  the  two  hyperbolas  are  said  to  be  conjugate  to  each 


Fig.  80. — A  Family  of  Conjugate 
Pairs  of  Hyperbolas  with  Common 
Asymptotes.  (An  interference  pat- 
tern made  from  a  glass  plate  under 
compression.  From  R.  Strauble, 
•'Ueber  die  Elasticitats-zahlen  und 
moduln  des  Glases."  Wied.  Ann. 
Bd.  68,  1899,  p.  38L) 


160        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§89 

other.  Thus  (1)  and  (5)  are  two  hyperbolas  which  are  conju- 
gate to  each  other.  Obviously  a  hyperbola  and  its  conjugate 
completely  bound  the  space  about  the  origin,  except  the  cuts  or 
lines  represented  by  the  common  asymptotes. 

Fig.  80  shows  a  family  of  pairs  of  conjugate  hyperbolas. 


Exercises 

1.  Sketch  on'the  same  pair  of 

axes  the  four  following  hyperbolas  and 

their  asymptotes: 

(1)  x^  -  i/2  =  25 

a;2       7/2 
(3)  25  -9=1 

(2)  a;2  -  1/2  =  -  25 

(4)  25-1  =  -  1- 

2.  Find  the  axes  of    the  hyperbola  y  =   ±  f  Vx^  -  64. 

3.  Compare  the  curves : 

X- 

-  ^'  =  1 
62        ^ 

a2 

and 

a:2 

y2 
_        *'.       —       _     1 

a2 

4.  Compare  the  curves: 

X2 

-!e- 

9 

and 

X2 

-I--- 

16 

5.  Write  the  equation  of  the 

hyperbola  conjugate  to 

y  = 

±  f  Va;2  -  64. 

6.  Compare  the  graphs  of : 

y  = 

+  f  Vx2  -  64 

y  = 

±  1  Vx2  -  16 

y  =  ±  I  Vx2  -  4 

y    =     ±    I  Vx2    -   1 

y   =    ±1  Vx2  -  1/16 

y   =    +   I  Vx2   -  0. 

7.  Show  that  3^2  -  4?/2  -  7a;  +  5i/  +  2  =  0  is  a  hyperbola.  Find 
the  position  of  the  center  and  of  the  vertices.  The  vertices  locate 
the  so-called  "limiting  lines"  of  the  hyperbola. 


§89]  THE  ELLIPSE  AND  HYPERBOLA  161 

8.  Show  that  x^  —  4x  —  4y^  +  42/  =  4  is  a  hyperbola.  Find  the 
limiting  lines  and  center. 

9.  Discuss  the  graphs: 

x^  -  y'^  =  1 
and 

2/2  -  x^  =  1. 

10.  Discuss  the  graph  IQx^  —  y^  —  40a;  —  Qy  =  2,  and  find  the 
limiting  lines. 

11.  In  Fig.  77,  show  that  DS  =  PD  and,  hence,  from  the  triangle 
DSO,  x2  -  2/2  =  a\ 

12.  In  Fig.  77,  show  that  PK  =  x  -  y,  PK'  =  x  +  y,  and  that  the 
rectangle  PK  X  PK'  is  constant  for  all  positions  of  P  and  equal  to  the 
square  on  OA. 


11 


CHAPTER  V 
SINGLE  AND  SIMULTANEOUS  EQUATIONS 

90.  The  Rational  Integral  Fvinction  of  x.  The  general  form  of 
a  polynomial  of  the  nth  degree  is: 

aoX""  +  flix'^-i  +  aiX""-^  +   •    •    .  +  an-ix  +  a„ 
where  the  symbols,  ao,  ai,  a2,   .    .    .   ,  stand  for  any  real  constants 
whatsoever,  positive  or  negative,  integral  or  fractional,  rational  or 
irrational,  and  where  n  is  any  positive  integer.     The  number  of 
terms  in  the  rational,  integral  function  of  the  nth  degree  is  (n  +  1). 

91.  The  Remainder  Theorem.  //  a  rational  integral  function  of 
X  he  divided  by  {x  —  r)  the  remainder  which  does  not  contain  x  is 
obtained  by  writing,  in  the  given  function,  r  in  place  of  x:  This 
theorem  means,  for  example  that  the  remainder  of  the  division: 

(x3  -  6x2  +  llx  -  6)  4-  (a;  -  4)  ig  43  _  6(4)2  ^  11(4)  _  6  or  6 
Also  that  the  remainder  of  the  division: 
.  (x^  -  6x2  _^  iia,  _  6)  -^  (x  +  1) 

(-1)3  -  6(-l)2  +  ll(-l)-6=  -  24 
The  theorem  enables  one  to  write  the  remainder  without  actually 
performing  the  division. 
To  prove  the  theorem,  let 

/(x)  =  aox"  +  aiX"-i  +  a2X''-^  +   .    .    .   +  an-ix  +  an    (1) 
and:  /(r)  =  ao^"  +  air'^-^  +  a2r'»-2  +    .    .    .    +  a„-ir  +  a„     (2) 
then: 
/(x)  -/(r)  =  ao{x»  -r")  +  ai(x"-i  -  r^-i)  +  .    .    . 

+  an-i(x  -  r)  (3) 
The  right  side  of  this  equation  is  made  up  of  a  series  of  terms  con- 
taining differences  of  like  powers  of  x  and  r,  and,  hence,  by  the 
well-known  theorem  in  factoring,^  each  binomial  term  is  exactly 
divisible  by  (x  —  r).     The  quotient  of  the  right  side  of  (3)  by 

^  See  Appendix. 

162 


§92]        SINGLE  AND  SIMULTANEOUS  EQUATIONS         163 

{x  —  r)  may  be  written  out  at  length,  but  it  is  sufficient  to 
abbreviate  it  by  the  symbol  Q(x)   and  write: 

fcM  .  Q(,)  (4) 


or: 


m  .  Q(,)  +  i(Ll  (5) 


Now  if  N  be  any  dividend,  D  any  divisor,  and  Q  the  quotient  and 
R  the  remainder,  then: 

N/D  =  Q-]-RID  (6) 

This  form  applied  to  (5)  shows  that  fir)  is  the  remainder  when 
fix)  is  divided  by  ix  —  r).  Thus  the  Remainder  Theorem  is 
established. 

92.  The  Factor  Theorem.  If  a  rational  integral  function  of 
X  becomes  zero  when  r  is  written  in  the  place  of  x,  {x  —  r)  is  a  fac- 
tor of  the  function:  This  means,  for  example,  that  if  3  be  substi- 
tuted for  X  in  the  function  x^  —  Gx^  +  llx  —  6  and  the  result 
33  _  6(3)2  _f_  11(3)  -6  =  0,  then  ix  -  3)  is  a  factor  of 
x^  -  6.^2  -f  llx  -  6. 

This  theorem  is  but  a  corollary  to  the  remainder  theorem. 
For  if  the  substitution  x  =  r  renders  the  function  zero,  the 
remainder  when  the  function  is  divided  by  ix  —  r)  is  zero,  and  the 
theorem  is  established. 

The  value  r  of  the  variable  x  that  causes  the  function  to  take 
on  the  value  zero  has  already  been  named  a  root  or  a  zero  of  the 
function.  The  factor  theorem  may,  therefore,  be  stated  in  the 
form:  A  rational  integral  function  of  the  variable  x  is  exactly 
divisible  by  ix  —  r)  where  r  is  any  root  of  the  function. 

The  familiar  method  of  solving  a  quadratic  equation  by  factor- 
ing is  nothing  but  a  special  case  of  the  present  theorem.     Thus  if: 

a;2  -  5a;  +  6  =  0 
then: 

ix  -  2)ix  -  3)  =  0 
and  the  roots  are  x  =  2  and  x  =  3.     The  numbers  2  and  3  are 
such  that  when  substituted  in  x^  —  5x  +  6  the  expression  is 
zero;  and  the  factors  of   the   expression  are  x  —  2  and  x  —  3 
by  the  factor  theorem, 


Exercises 

1.  Tabulating 
obtain: 

the 

cubic 

polynomial 

X        -  3 

-  2 

-1 

-01        1 

164        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§93 


2        2.5        3    4 

fix),  -  120,  -  60,  -  24,  -  6,  0,  +  0.375,  0,  -  0.375,  0,  6 
What  is  the  remainder  when  the  function  is  divided  by  x  —  4? 
By  re +  2?     By  rp  +  3?     By  a;  -  1.5?     By  x  -  3? 
Name  three  factors  of  the  above  function. 

2.  Find  the  remainder  when  x'*  —  5x^  +  12x2  +  4x  —  8  is  divided 
by  X  -  2. 

3.  Show  by  the  remainder  theorem  that  x**  +  a""  is  divisible  by 
X  -\-  a  when  n  is  an  odd  integer,  but  that  the  remainder  is  2a"  when  n 
is  an  even  integer. 

4.  Without  actual  division,  show  that  x*  —  4x2  —  7a;  —  24  is 
divisible  by  x  —  3. 

5.  Show  that  a^  -\-  a^  —  a¥  —  b^  is  divisible  by  a  —  6, 

6.  Show  that  (6  -  c)(b  +  c)2  +  (c  -  a){c+  a)^  +  (a  -  b){a  +  6)2 
is  divisible  by  (6  —  c){c  —  a)  {a  —  h). 

7.  Show  that  (x  +  l)2(x  -  2)  -  4(x  -  l)(x  -  5)  +  4  is  divisible 
by  X  —  1. 

8.  Show  that  {b  -  c)^  +  {c  -  a)^  +  {a  -  b)^  is  divisible  by 
(6  —  c){c  —  a){a  —  b). 

9.  Show  that  6x6  _  3^4  _  53.3  ^  ^^^2  _  2x  -  3  is  divisible  by 
X  +  1. 

93.  It  follows  at  once  from  the  factor  theorem  that  it  is  possible 
to  set  up  an  equation  with  any  roots  desired;  for  example,  if  we 
desire  an  equation  with  the  roots  1,  2,  3  we  have  merely  to 
write: 

(x  -  l)(x  -  2)(x  -  3)  =  0  (1) 

Forming  the  product: 

x^  -  6a;2  +  llx  -  6  =  0 
or  transposing  the  terms  in  any  manner,  as : 
x3  +  llx  =  6x2  _|.  Q 

in  no  way  essentially  modifies  the  equation.  If,  however,  the 
equation  (1)  be  multiplied  through  by  any  function  of  x,  the 
number  of  roots  of  the  equation  may  be  increased.  Thus,  mul- 
tiplying (1)  by  (a;  +  2)  introduces  a  new  root  x  =  —2.    Likewise, 


§93]        SINGLE  AND  SIMULTANEOUS  EQUATIONS         166 

dividing  equation  (1)  through  by  the  factor  {x  —  2),  leaves  an 
equation: 

{x  -  i){x  -  3)  =  0  (2) 

which  lacks  the  root  x  =  2. 

By  the  principles  or  axioms  of  algebra,  an  equation  remains 
true  if  we  unite  the  same  number  to  both  sides  by  addition  or 
subtraction;  or  if  we  multiply  or  divide  both  members  by  the 
same  number,  not  zero;  or  if  like  powers  or  roots  of  both 
members  be  taken.  But  we  have  given  sufficient  illustrations  to 
show  that  these  operations  may  affect  the  number  of  roots  of  the 
equation.  This  is  obvious  enough  in  the  cases  already  cited. 
Sometimes,  however,  the  operation  that  removes  or  introduces 
a  root  is  so  natural  and  its  effect  is  so  disguised  that  the  student 
is  not  apt  to  take  due  account  of  its  effect.     Thus,  the  roots  of: 

3(x  -  5)  =  x{x  -  5)  +  x2  -  25  (3) 

are  —  1  and  5,  for  either  of  these  when  substituted  for  x  will 
satisfy  the  equation.  Dividing  the  equation  through  by  x  —  5, 
the  resulting  equation  is : 

3  =  X  +  X  +  5  (4) 

This  equation  is  not  satisfied  hjx  =  5.  One  root  has  disappeared 
in  the  transformation.  This  is  easy  to  keep  account  of  if  (3) 
be  given  in  the  form: 

(x  -  b)(x  +  1)  =  0,  (5) 

but  the  fact  that  a  factor  has  been  removed  may  be  overlooked 
when  the  equation  is  written  in  the  form  first  given. 

A  very  important  effect  upon  the  roots  of  an  equation  results 
from  squaring  both  members.  The  student  must  always  take 
proper  account  of  the  effect  of  this  common  operation.  To  il- 
lustrate, take  the  equation: 

X  +  5  =  1  -  2x  (6) 

It  is  satisfied  only  by  the  value  x  =  —  4/3.  Now,  by  squaring 
both  sides  of  the  equation,  we  obtain: 

a:2  +  lOx  +  25  =  1  -  4x  +  4^2  (7) 

which  is  satisfied  by  either  a;  =  6ora;=  — 4/3.  Here,  obviously, 
an  extraneous  solution  has  been  introduced  by  the  operation  of 
squaring  both  members. 


166        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§93 

It  is  easy  to  show  that  squaring  both  members  of  an  equation 
is  equivalent  to  multiplying  both  sides  by  the  sum  of  the  left  and 
right  members.     Thus,  let  any  equation  be  represented  by: 

L{x)  =  R{x)  (8) 

in  which  L(x)  represents  the  given  function  of  x  that  stands  on 
the  left  side  of  the  equation  Siii.dR{x)  represents  the  given  function 
of  X  that  stands  on  the  right  side  of  the  equation. 
Squaring  both  sides: 

[L(x)]2  =  [R{x)V  (9) 

Transposing : 

[L{x)Y  -  [R{x)Y  =  0  (10) 

or  factoring; 

[L{x)  +  R{x)]  [L{x)  -  R{x)]  =  0  (11) 

But  (8)  may  be  written: 

L{x)  -  Rix)  =  0  (12) 

Thus,   by  squaring  the  members   of  the  equation  the  factor 
L{x)  +  R{x)  has  been  introduced. 

The  sum  of  the  left  and  right  members  of  (6),  above,  is  6  —  x. 
Hence,  squaring  both  sides  of  (6)  is  equivalent  to  the  introduction 
of  this  factor,  or,  the  operation  introduces  the  root  6,  as  already 
noted. 

As  another  example,  suppose  that  it  is  required  to  solve: 

sin  q:  cos  CK  =  1  /4  (13) 

for  a  <  90°.     Substituting  for  cos  a: 


sin  a  VI  —  sin^  a  = 

=  1/4 

(14) 

squaring: 

sin^  a  {1  —  sin^  a)  = 

=  1/16 

(15) 

completing  the  square: 

sin^  a  —  sin^  a  +  1  /4  = 
Hence: 

=  3/16 

(16) 

sin  a  =  ±  Vl/2  +  (1/4)^3 

=  ±  0.9659  or  ±  0.2588  (17) 

Only  the  positive  values  satisfy  (13);  the  negative  values  were 
introduced  in  squaring  (14).  If,  however,  the  restriction  a  <  90° 
be  removed,  so  that  the  radical  in  (14)  must  be  written  with  the  double 
sign,  then  no  new  solutions  are  introduced  by  squaring. 


§94]         SINGLE  AND  SIMULTANEOUS  EQUATIONS         167 

94.  Legitimate   and   Questionable   Transformations.     If   one 

equation  is  derived  from  another  by  an  operation  which  has  no 
effect  one  way  or  another  on  the  solution,  it  is  spoken  of  as  a 
legitimate  transformation;  if  the  operation  does  have  an  effect 
upon  the  final  result,  it  is  called  a  questionable  transformation, 
meaning  thereby  that  the  effect  of  the  operation  requires  ex- 
amination. 

In  performing  operations  on  the  members  of  equations,  the 
effect  on  the  solution  must  be  noted,  and  proper  allowance 
made  in  the  result.  It  cannot  be  too  strongly  emphasized  that 
the  test  for  any  solution  of  an  equation  is  that  it  satisfy  the  original 
equation.  ''No  matter  how  elaborate  or  ingenious  the  process 
by  which  the  solution  has  been  obtained,  if  it  do  not  stand  this 
test  it  is  no  solution;  and,  on  the  other  hand,  no  matter  how  simply 
obtained,  provided  it  do  stand  this  test,  it  is  a  solution."^ 

Among  the  common  operations  that  have  no  effect  on  the  solu- 
tion are  multiplication  or  division  by  known  numbers,  or  addition 
or  subtraction  of  like  terms  to  both  members ;  none  of  these  intro- 
duce factors  containing  the  unknown  number.  Taking  the 
square  root  of  both  numbers  is  legitimate  if  the  double  sign  be 
given  to  the  radical.  Clearing  of  fractions  is  legitimate  if  it  be  done 
so  as  not  to  introduce  a  new  factor.  If  the  fractions  are  not  in 
their  lowest  terms,  or  if  the  equation  be  multiplied  through  by  an 
expression  having  more  factors  than  the  least  common  multiple 
of  the  denominators,  new  solutions  may  appear,  for  extra  factors 
are  probably  thereby  introduced.  Hence,  in  clearing  of  fractions 
the  multiplier  should  be  the  least  common  denominator  and  the 
fractions  should  be  in  their  lowest  terms.  This,  however,  does  not 
constitute  a  sufficient  condition,  therefore  the  only  certainty  lies 
in  checking  all  results. 

Exercises 

Suggestions:  It  is  important  to  know  that  any  equation  of 
the  form 

ax^"  -f-  6x"  -h  c  =  0 
can  be  solved  as  a  quadratic  by  finding  the  two  values  of  x". 
Frequently  equations  of  this  type  appear  in  the  form 
dx"*  -{-  ex-"  =  f 

»  Chrystal's  Algebra. 


168        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§94 

Likewise  any  equation  of  the  form 

af{x)  +  b  ylfix)  +  c  =  0 
can  be  solved  as  a  quadratic  by  finding  the  two  values  of  V/(x) 
and  then  solving  the  two  equations  resulting  from  putting  ^lf{x) 
equal  to  each  of  them.     One  of  these  usually  gives  extraneous 
solutions. 

These  two  types  occur  in  the  exercises  given  below. 

Since  operations  which  introduce  extraneous  solutions  are 
often  used  in  solving  equations,  the  only  sure  test  for  the  solution 
of  any  equation  is  to  check  the  results  by  substituting  them  in 
the  original  equation. 

Take  account  of  all  questionable  operations  in  solving  the  following 
equations : 

Sx 6_  9 

'  X  -S  ~  x-\-S'^  X  -  S' 

2.  (x2  +  5x+Q)/{x  -  3)  +  4x  -.  7  =   -  15. 

3.  3(x  -  5){x  -  l)ix  -2)  =  {x  -  b){x  +  2){x  +  3). 
Note:     Divide  by  {x  —  5),  but  take  account  of  its  effect. 

4.  x^/a  -\-  ax  =  x^/h  +  hx. 

5.  ax{cx  —  35)  =  5a (36  —  ex). 

6.  x2  -  n2  =  n  -  x. 

7.  {X  -  4)3  +  (a;  -  5)^  =  3l[(x  -  4)2  -  {x  -  5)^].       Divide  .  by 
{x-  4:)  +  {x-b)  or  2x  -  9. 

x^  —  Zx  1 

8.  -^ \ — h  2  -1 ^  =  0.     If  the  fractions  be  added,  multi- 

x^  —  \  X  —  1  ' 

plication  is  unnecessary.     There  is  only  one  root. 

9.  a;  =  7  -  Va;2  -  7. 

10.  Vx  +  20  -  V^^^  -3=0. 

11.  \/15/4  +  x  =  3/2  +  V^ 


12.  20x/Vl0x  -  9  -  VlOx  -  9  =  18/VlOx  -9+9. 


13.  — = =  =  — — i^  •       Consider  as  a  proportion  and  take 

Vx   -  Vx-Z       ^-^ 

by  composition  and  division. 

14.  x^^  +  5/2  =  (13/4)x''^. 

15.  \/^  -  2V^+  a;  =0.     Divide  by  Vx . 

16.  2Va:2-  5x  +  2  -  a;^  +  gx  =  3x  -  6.     Call  x"^  -  bx  ■\- 2  =  u\ 


§95]         SINGLE  AND  SIMULTANEOUS  EQUATIONS         169 


17.  4x2  -  4x  +  20  V2  a;2  -  5  X  +  6  =  6x  +  66. 

18.  x-2  -  2x-i  =  8. 

19.  x^^»  -  Sx''''^^  +4=0. 

20.  llOx-4  +  1  =  21x-2. 

21.  Vx  +  4x->'^  =  5. 

22.  Sx^^  -Sx-'^  =  63. 

23.  (x  -  a)"  -  3 (x  -  a)-«  =  2. 

24.  2x^  -  3x^^  +  x  =  0. 

95.  Intersection  of  Loci.  Any  pair  of  values  of  x  and  y  that 
satisfies  an  equation  containing  x  and  y  locates  some  point  on 
the  graph  of  that  equation.  Consequently,  any  set  of  values  of 
X  and  y  that  satisfies  both  equations  of  a  system  of  two  equations 
containing  x  and  y,  must  locate  some  point  common  to  the 
graphs  of  the  two  equations.  In  other  words,  the  coordinates 
of  a  point  of  intersection  of  two  graphs  is  a  solution  of  the  equations 
of  the  graphs  considered  as  simultaneous  equations. 

To  find  the  values  of  x  and  y  that  satisfy  two  equations,  we 
solve  them  as  simultaneous  equations.  Hence,  to  find  the  points 
of  intersection  of  two  loci  we  must  solve  the  equations  of  the 
two  curves.  There  will  be  a  pair  of  values  or  a  solution  for  each 
point  of  intersection. 

Thus,  the  intersection  of  the  lines  y  =  Sx  -  2  and  ?/  =  x  /2  +  3 
is  the  point  (2,  4)  and  x  =  2,  y  =  4,  is  the  solution  of  the  simul- 
taneous equations. 

To  find  the  points  of  intersection  of  the  circle  x^  +  y'^  =  25 
and  the  straight  line  x  -\-  y  =  9  we  solve  the  equations  by  the 
usual  method,  as  follows: 

x'-{-y'  =25)  (1) 

X   -{-y    =    7!  (2) 

The  graphs  are  a  straight  line  and  a  circle,  as  shown  in  (1), 
Fig.  81.     Squaring  the  second  equation,  the  system  becomes: 

x2  +  t/2  =  25  \  (3) 

x2  +  2xy  +  ?/2  =  49  /  C4) 

The  second  equation  represents  the  two  straight  lines  shown  in 
Fig.  81,  (2).  The  effect  of  squaring  has  been  to  introduce  two 
extraneous  solutions  corresponding  to  the  points  Pz  and  P4. 


170        ELEMENTARY  MATHEMATICAL  ANALYSIS 


>95 


Multiplying  (3)  by  2  and  subtracting  (4)  from  it,  the  last  pair 
of  equations  becomes: 

x^-2xy  +  y^=    l\  (6) 

x2  +  2xy  +  2/2  =  49  j  (7) 

which  gives  the  four  straight  lines  of  Fig.  81,  (4).     Taking  the 

square  root  of  each  member,  but  discarding  the  equation  x  -\-  y  = 

—  7,  because  it  corre- 
sponds to  the  extraneous 
solutions  introduced  by 
the  questionable  operation, 
we  have: 

x-y  =  ±1\       (8) 
x  +  y  =  7       I       (9) 
By  addition  and  subtrac- 
tion we  obtain  the  results : 


x  =  S\ 

2/  =  4/ 

a;  =  41 

2/  =  3/ 


(10) 


(11) 


Fig.  81. — Graphic  Representation  of 
the  Steps  in  the  Solution  of  a  Certain  set 
of  Simultaneous  Equations. 


represented  by  the  inter- 
sections of  the  lines  parallel 
to  the  axes  shown  in  Fig. 
81,  (5).  ^ 

This  is  a  good  illustra- 
tion of  the  graphical 
changes  that  take  place 
during  the  solution  of  sim- 
ultaneous equations  of  the 
second  degree.  The  ordinary  algebraic  solution  consists,  geo- 
metrically, in  the  successive  replacement  of  loci  by  others  of  an 
entirely  different  kind,  but  all  passing  through  the  points  of  in- 
tersection (as  Pi,  Ps,  Fig.  81)  of  the  original  loci. 

Exercises 
1.  Find  the  points  of  intersection  of  the  circle  and  parabola: 

rc2  +  2/2  =  5 
7/2  =  4x. 


§96]         SINGLE  AND  SIMULTANEOUS  EQUATIONS         171 

Note  that  of  the  two  lines  parallel  to  the  y-axis,  given  by  the  equation 
x2  +  4x  —  5  =  0,  one  does  not  cut  the  circle:  x^  -\-  y"^  =  5. 

2.  Find  the  points  of  intersection  of  x^  +  i/^  =  5  and  the  hyperbola 

3.  Solve,  by  graphical  means  only,  to  two  decimal  places : 

y  =  x^  -\-  X  —  \ 
xy  =  1. 

4.  Solve  in  like  manner : 

a;2  +  2/2  =  16 
x^  -  2xy  +2/2  =  9. 
Reason  out  what. each  equation  represents    before  attempting  to 
graph. 

5.  Solve  in  like  manner : 

x^+y^+x-\-y  =  7 
2x2  +  2^/2  -  4x  +  4?/  =  8. 
These  loci  should  be  graphed  without  tabulating  numerical  values 
of  the  variables. 

6.  Solve  graphically: 

W2    +  „2    =    9 
^2    _   y2    =    4, 

Note.  Draw  the  lines  x  +  y  =  9,  and  x  —  y  =  A.  The  values 
of  X  and  y  determined  by  the  intersection  of  these  lines  are  the 
values  of  u^  and  v^  respectively,  from  which  u  and  v  can  be  computed. 

7.  Solve  the  system : 

a;2  4.  ^2  =  10 

xym  +  2/2/9  =  1. 
96.  Quadratic    Systems.  ^    Any    linear-quadratic    system    of 
simultaneous  equations,  such  as: 

y  =  mx  +  k 
ax^  4-  hy^  +  2hxy  +  2gx  +  2fy  +  c  =  0 
can  always  be  solved  analytically;  for  y  may  readily  be  eliminated 
by  substituting  from  the  first  equation  into  the  second.  A 
system  of  two  quadratic  equations  may,  however,  lead,  after 
elimination,  to  an  equation  of  the  third  or  fourth  degree;  and, 
hence,  such  equations  cannot,  in  general,  be  solved  until  the 
solutions  of  the  cubic  and  biquadratic  equations  have  been 
explained. 

1  A  large  part  of  the  remainder  of  this  chapter  can  be  oniitted  if  the  students 
have  had  a  good  course  in  algebra  in  the  secondary  school. 


172        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§97 

A  single  illustration  will  show  that  an  equation  of  the  fourth 
degree  may  result  from  the  elimination  of  an  unknown  number 
between  two  quadratics.     Thus,  let: 

x2  _    ^  ==    5 

x^  -\-  xy  =  10 
From  the  first,  y  =  x"^  —  5.     Substituting  this  value  of  y  in  the 
second  equation,  and  performing  the  indicated  operations,  we 
obtain : 

x^  +  x^  -  5x  +  10  =  0. 
While,  in  general,  a  bi-quadratic  equation  results  from  the 
process  of  elimination  from  two  quadratic  equations,  there  are 
special  cases  of  some  importance  in  which  the  resulting  equation 
is  either  a  quadratic  equation  or  a  higher  equation  in  the  quadratic 
form.     Two  of  these  cases  are: 

(1)  Systems  in  which  the  terms  containing  the  unknown  num- 
bers are  homogeneous;  that  is,  systems  in  which  the  terms  con- 
taining the  unknown  numbers  are  all  of  the  second  degree  with 
respect  to  the  unknown  numbers,  such,  for  example,  as: 

x^  —  2xy   =    5 
3a;2  -  lOt/2  =  35 

(2)  Systems  in  which  both  equations  are  symmetrical;  that  is, 
such  that  interchanging  x  and  y  in  every  term  does  not  alter  the 
equations;  for  example: 

x^  +  y^  -  X  -  y  =  7S 
xy  -\-  X  -\-  y  =  39 
97.  Unknown    Terms    Homogeneous.     The    following    work 
illustrates  the  reasoning  that  will  lead  to  a  solution  when  applied 
to  any  quadratic  system  all  of  whose  terms  containing  x  and  y 
are  of  the  second  degree.    Let  the  system  be: 

x^  —  xy  =  2 
2x2  +  y^  =  Q  (1) 

Divide  each  through  by  x'^  (or  y"^),  then: 

1-    iy/x)  =  2/x' 

2  +  {ylxy  =  9lx^  (2) 

Since  the  left  members  were  homogeneous,  dividing  by  x^  renders 
them  functions  of  the  ratio  {y  /x)  alone;  call  this  ratio  m.     Then 


§97]        SINGLE  AND  SIMULTANEOUS  EQUATIONS         173 

equations  (2)  contain  only  the  unknown  numbers  m  and  x^. 
The  latter  is  readily  eliminated  by  subtraction,  leaving  a  quad- 
ratic for  the  determination  of  m.  When  m  is  known,  substituting 
in  (2)  determines  x,  and  the  relation  y  =  mx  determines  the 
corresponding  values  of  y. 

The  above  illustrates  the  principles  on  which  the  solution  is 
based.  In  practice,  it  is  usual  to  substitute  y  =  mx  at  once,  and 
then  eliminate  x^  by  comparison;  thus,  from  the  substitution 
y  =  mx  in  (1),  we  obtain: 

x^  —  mx^  =  2 
2a;2  +  m^x^  =  9  (3) 


Thence: 

Whence: 
or: 

Factoring : 
whence: 
Hence: 


x^  =  2/(1  -  m) 

x2  =  9/(2  +  m2)  (4) 

2/(1  -  m)  =  9/(2  +  m2)  (5) 

2m2  +  9m  =  5  (6) 

(2m  -  1)  (m  +  5)  =  0  (7) 

m  =  1  /2  or  -  5  (8) 


x  =  +  2or  +  (l/3)\/3 
2/=  +  lor  +  (5/3)V'3  (9) 

These  solutions  should  be  written  as  corresponding  pairs  of  values 
as  follows: 

x  =  2         x=-2         x=       (l/3)\/3  x=-(l/3)\A3 

y  =  l         y=-l         2/=-(5/3)\/3  y=       (5/3)\/3 

This  system  can  readily  be  solved  without  the  use  of  the  mx 
substitution  by  merely  solving  the  first  equation  for  y  and  sub- 
stituting in  the  second. 

Graphically  (see  Fig.  82),  the  above  problem  is  equivalent  to 
finding  the  intersections  of  the  curves: 

_x{x  -  y)  =  2 
(V2a:)2  +  2/2  =  9 

The  first  is  a  curve  with  the  two  asymptotes  a;  =  0  and  x  —  y 


174         ELEMENTARY  MATHEMATICAL  ANALYSIS 


=  0.  As  a  matter  of  fact,  the  curve  is  a  hyperbola,  although  proof 
that  such  is  the  case  cannot  be  given  until  the  method  of  rotating 
any  curve  about  the  origin  has  been  explained.  The  second  curve 
is  obviously  an  ellipse  generated  from  a  circle  of  radius  3  by 
shortening  the  abscissas  in  the  ratio  v/2:l.  The  two  curves 
intersect  at  the  points: 

X  =  2  -  2  0.557   ...  -  0.557   .  .  . 

y  =  1  -  1  -  2.887  ...  +  2.887   . . . 


\ 
\ 

\ 

Y 

\ 

\ 

\ 

i 

4 
3 

// 

■  (h 

2       \ 
1       / 

'                              X 

-4       -3        -2/-\^^^ 

O     1  / 

V  / 

\     / 
\   / 

2         3         4 

/ 

~  X  .  1/3  VT~ 

'W~~ 

-  i/  =  -   6/8  WT 

Y 

\ 

\ 

\ 

Fig.  82. — Solutions  of  a  Set  of  Simultaneous  Quadratics  given  graph- 
ically by  the  coordinates  of  the  points  of  Intersection  of  the  Ellipse  and 
Hyperbola. 


The  auxiliary  lines,  y  =  \x  and  y  =  —  5x,  made  use  of  in  the 
solution  are  shown  by  the  dotted  lines. 

98.  Sjrmmetrical  Systems.  Simultaneous  quadratics  of  this 
type  are  always  readily  solved  analytically  by  seeking  for  the  values 
of  the  binomials  x  +  y  md  x  —  y.     The  ingenuity  of  the  student 


SINGLE  AND  SIMULTANEOUS  EQUATIONS         175 

will  usually  show  many  short  cuts  or  special  expedients  adapted 
to  the  particular  problem.  The  following  worked  examples 
point  out  some  of  the  more  common  artifices  used. 

1.  Solve 

x  +  y  =  Q  (1) 

xy  =  5  (2) 

Squaring  (1) 

X'  +  2xy  +  2/2  =  36  (3) 

Subtracting  four  times  (2)  from  (3) : 

x^  -  2xy  +  2/^^  =  16 
whence: 

But  from  (1): 

Therefore: 

2.  Solve 


X  —  y  =  +4 
X  -\-  y  =  Q 

X  =  5  X  =  1 

y  =  I  y  =  5 


x^  +  y^  =  34  (1) 

xy  =  15  (2) 

Adding  two  times  (2)  to  (1): 

x'  -f  2xy  4-  y^  =  64  (3) 

Subtracting  two  times  (2)  from  (1): 

x2  -  2x2/  4-  ?/2  =  4  (4) 

Whence,  from  (3)  and  (4) : 

x  +  y  =  ±8 
X  - y  =  ±2 
Therefore : 

x  =  5.  X  =  S  X  =  —  5  X  =  —  3 

y  =  S  2/  =  5  y  =  -  3  2/ =-5 

The  hyperbola  and  circle  represented  by  (1)  and  (2)  should  be 
drawn  by  the  student. 

3- 

x^  +  y^  =  72  (1) 

X   +y    =    ^  (2) 

Cubing  (2) :  ' 

x^  +  Sx^y  +  3xy^  +  2/'  =  216  (3) 


176        ELEMENTARY  MATHEMATICAL  ANALYSIS        [§98 

Subtracting  (1)  and  dividing  by  3: 

xy{x  +  2/)  =  48  (4) 

whence,  since 

X  +  y  =  Q 
we  have  xy  =  8  (5) 

From  (2)  and  (5)  proceed  as  in  example  1,  and  find: 
X  =  4t  X  =  2 

2/  =  2  2/  =  4 

Otherwise,  divide  (1)  by  (2)  and  proceed  by  the  usual  method. 
4.  Solve 

x^  +  xy  =  (7/3)(x  +  2/)  (1) 

y'^  +  xy  =  (ll/3)(x  +  2/)  (2) 

adding  (1)  and  (2): 

{x  +  yy  -Q{x  +  y)  =0  (3) 

whence: 

x-\-y  =  OotQ  (4) 

Now,  because  x  +  ?/  is  a  factor  of  both  members  of  (1)  and  (2), 
the  original  equations  are  satisfied  by  the  unlimited  number  of 
pairs  of  values  of  x  and  y  whose  sum  is  zero,  namely,  the  coor- 
dinates of  all  points  on  the  line  x  -\-  y  =  0. 
Dividing  (1)  by  (2),  we  get: 

xfy  =  7in 
This,  and  the  fine  x  +  y  =  Q,  from  (4),  give  the  solution: 

X  =  7/3 
y  =  11/3 
Graphically,  the  equation  (1)  is  the  two  straight  lines: 

(x  -  7 IS){x -\- y)  =0 
Equation  (2)  is  the  two  straight  lines: 

{y-  ni3)ix  +  y)  =0 
These  loci  intersect  in  the  point  (7/3,  11/3)  and  also  intersect 
everywhere  on  the  line  x  -^  y  =  0. 

Exercises 
1.  Show  that: 

rc2  +  ^2  =  25 

X  +  y  =  1 


§99]        SINGLE  AND  SIMULTANEOUS  EQUATIONS         177 

has  a  solution,  but  that  there  is  no  real  solution  of  the  system: 
a;2  +  2/2  =  25 
x+y  =  11. 

2.  Do  the  curves : 

x2  +  2/2  =  25 

xy  =  100,  intersect? 
Do  the  curves :  x^  +  y^  =  25 

xy  =  12,  intersect? 

3.  Solve: 

(x2  +y^){x  +  y)  =272 
x^  +  y^  -\-  x  +  y  =  ^2. 
Note:     Call  x^  -{-  y^  =  u,  and  x  -{-  y  =  v. 

4.  Show  that  there  are  four  real  solutions  to: 

X2    +   1/2    -    12    =   X    +  1/  , 

o:?/  +  8  =  2(x  +  y). 

5.  Solve:  x"^  -\-  y"^  +  x  -\- y  =  1% 

xy  =  6. 

99.  Graphical  Solution  of  the  Cubic  Equation.  The  roots  of  a 
cubic  x^  +  cLx"^  -\-  hx  +  c  =  0  (where  a,  h,  and  c  are  given  known 
numbers)  may  be  determined  graphically  as  explained  in  §39, 
or  we  may  proceed  as  follows:  The  next  highest  term  in  the 
cubic  may  be  removed  by  the  substitution  x  =  Xi  —  ajZ,  as  may 
readily  be  shown  by  trial.  Hence,  it  is  merely  necessary  to  con- 
sider cubic  equations  of  the  form: 

x^  -\-  ax-\-h  =  Q  (1) 

Consider  the  system  of  equations: 

1/  =  x»  1 

y  =  —  ax  —  0    J 

Graphically,  the  curves  consist  of  the  cubic  parabola  (Fig.  83) 
and  a  straight  line.  The  intersections  of  the  two  graphs  give  the 
solutions  of  the  system.     Eliminating  y  by  subtraction,  we  obtain 

x^  +  ax  -\-h  =  0 

which  shows  that  the  values  of  x  that  satisfy  the  system  (2) 
give  the  roots  of  equation  (1).  Hence  (1)  may  be  solved  by  means 
of  the  graph  of  (2).  In  this  graph  the  cubic  parabola  y  =  x^ 
is  the  same  for  all  cubics;  hence  if  the  cubic  parabola  be  once 
drawn  accurately  to  scale,  then  all  cubic  equations  can  be  solved 

12 


178        ELEMENTARY  MATHEMATICAL  ANALYSIS 


m 


by  properly  drawing  the  appropriate  straight  line,  or  by  properly 
laying  a  straight  edge  across  the  graph  of  the  cubic  parabola. 

In  drawing  the  graph  of  the  cubic  parabola,  it  is  desirable  to 
use,  for  the  ?/-scale,  one-tenth  of  the  unit  used  for  the  x-scale,  so  as 
to  bring  a  greater  range  of  values  for  y  upon  an  ordinary  sheet  of 

coordinate  paper.  The  cubic  parab- 
ola graphed  to  this  scale  is  shown 
in  Fig.  83.  The  diagram  gives  the 
solution  of  rc^  —  a:  —  1  =  0.  The 
graphs  y  =  x^  and  y  =  x  -^  1  are 
seen  to  intersect  at  x  =  1.32.  This, 
then,  should  be  one  root  of  the 
cubic  correct  to  two  decimal  places. 
The  line  y  =  x  -\-  1  cuts  the  cubic 
parabola  in  but  one  point,  which 
shows  that  there  is  but  one  real  root 
of  the  cubic.  To  obtain  the  imagi- 
nary roots,  divide  x^  —  x  —  1  by 
X  -  1.32.  The  result  of  the  divi- 
sion, retaining  but  two  places  of 
decimals  in  the  coefficients,  is: 

x^  +  1.32a;  +  0.7424  (3) 

Putting  this  equal  to  zero  and  solv- 
ing by  completing  the  square,  we 
find: 


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tf 

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Fig.  83.— A  Graphical 
Scheme  for  the  Solution  of 
Cubic  Equations. 


rc  =  -  0.66  ±  V-  0.3068 


or: 


x=  -  0.66  ±  0.55\/-  1  (4) 

in  which,  of  course,  the  coefficients  are  not  correct  to  more  than 
two  places. 
The  equation: 

x3  -  10a;  -  10  =  0  (5) 

illustrates  a  case  in  which  the  cubic  has  three  real  roots.  The 
straight  line  y  =  lOx  -f-  10  cuts  the  cubic  parabola  (see  Fig.  83) 
at  X  =  -  1.2,  a;  =  —  2.4,  and  x  =  3.6.  These,  then,  are  the 
approximate  roots.     The  product : 

{x  -h  1.2)(x  -h  2.4)(a;  -  3.6)  =  x^  -  10.08x  -  10.37 


§99]         SINGLE  AND  SIMULTANEOUS  EQUATIONS         179 

should  give  the  original  equation  (5).    This  result  checks  the  work 
to  about  two  decimal  places. 

It  is  obvious  that  a  similar  process  will  apply  to  any  equation 
of  the  form 

X"  -{-  ax  -{-  b  =  0 

The  x-scale  of  Fig.  83  extends  only  from  —  5  to  +5.  The 
same  diagram  may,  however,  be  used  for  any  range  of  values  by 
suitably  changing  the  unit  of  measure  on  the  two  scales;  thus,  the 
divisions  of  the  x-scale  may  be  marked  with  numbers  5-fold  the 
present  numbers,  in  which  case  the  numbers  on  the  y-scale  must  be 
marked  with  numbers  125  times  as  great  as  the  present  numbers. 
These  results  are  shown  by  the  auxiliary  numbers  attached  to 
the  y-scsile  in  Fig.  83. ^ 

Exercises 

Solve  graphically  the  following  equations  checking  each  result 
separately. 

1.  x^  -  Ax  +  10  =  0. 

2.  a;3  -12x  -  8  =  0. 

3.  a:3  +      .T  -  3  =  0. 

4.  x^  -15x  -5=0. 

5.  a;3  -  3x  +  1  =  0. 

6.  a:3  -  4x  -  2  =  0. 

7.  2  sin  0  +  3  cos  d  =  1.5. 

Note:  Construct  on  polar  paper  the  circles  p  =  2  sin  d  and 
p  =  3  cos  9. 

8.  2x  +  sin  a;  =  0.6. 

Note:  Find  the  intersection  oi  y  =  sin  x  and  the  line 
y  =  —  2x  -\-  0.6.  If  1.15  inches  is  the  amplitude  of  y  =  sin  x, then 
1.15  must  be  the  unit  of  measure  used  for  the  construction  of  the 
line  y  =  -  2x  +  0.6. 

9.  x^  +  X  +  1  +  1/x  =  0. 

10.  Show  that  x^  +  ax  +  b  =0  can  have  but  one  real  root  if  a  >  0. 

11.  (a)  Show  that  the  graph  oi  y  =  x^  +  bx  is  symmetrical  with 
respect  to  the  origin.     (See  §37,  equation  (1).) 

iFor  other  graphical  methods  of  solution  of  equaitons,  see  Runge's  "  Graph- 
ical Methods,"  Columbia  University  Press,  1912. 


180        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§100 

(6)  Show  that  the  graph  oi  y  =  x^  -j-  bx  -\-  c  is  symmetrical  with 
respect  to  the  point  (0,  c). 

(c)  If  the  substitution  x  =  Xi  —  a/3  removes  the  term  ax^  from  the 
equation  y  =  x^  -\-  ax^  +  6x  +  c,  show  thafc  the  graph  of  this  last 
equation  must  be  symmetrical  with  respect  to  some  point. 

12.  On  polar  paper,  draw  a  curve  showing  the  variation  of  local  or 
mean  solar  time  with  the  longitude  of  points  on  the  earth's  surface. 

If  it  be  noon  by  both  standard  and  mean  solar  (local)  time  at  Green- 
wich, longitude  0°,  construct  a  graph  on  polar  paper  showing  standard 
time  at  all  other  longitudes,  if  the  longitude  of  a  point  be  represented 
by  the  vectorial  angle  on  polar  paper  and  if  time  relative  to  Greenwich 
be  represented  on  the  radius  vector  using  1  cm.  =2  hours,  and  also  if 
it  be  assumed  that  the  changes  of  standard  time  take  place  exactly  at 
15°  intervals  beginning  at  7|°  west  longitude. 

If  it  be  noon  at  Greenwich,  write  an  equation  which  will  express  the 
local  time  of  any  point  in  terms  of  the  longitude  of  the  point.  Does  the 
expression  hold  for  points  having  negative  longitude?  Does  this 
function  possess  a  discontinuity? 

Can  a  similar  expression  be  written  giving  the  standard  time  at  any 
point  in  terms  of  the  longitude  of  the  point? 

If  t  be  standard  time  and  6  longitude,  and  if  the  functional  relation 
by  expressed  by  /,  so  that : 

t  =  m 

is  /  a  continuous  or  a  discontinuous  function?     Is  the  function  / 
defined  for  6  =  15°,  30°,  45°,  etc.,  and  why? 
In  actual  practice,  how  is  the  function  /  given? 

100.  Method  of  Successive  Approximations.  The  graphic 
method  of  solving  numerical  equations,  combined  with  the  method 
explained  below,  is  the  only  method  which  is  universally  ap- 
plicable. It  therefore  possesses  a  practical  importance  exceeding 
that  of  any  other  method.     An  example  will  illustrate  the  method. 

Suppose  that  it  is  required  to  find  to  four  decimal  places  one 
root  of  x3  -  X  -  1  =  0.  See  §99  and  Fig.  83.  The  graphic 
method  gives  x  =  1.32.  This  is  the  first  approximation.  A 
second  approximation  is  found  as  follows:  Build  the  table 
of  values  fory  =  x^  —  x  —  l 

y 


1.32 
1.33 


0.01 


-  .0200 
+  .0226 


.0426    Differences. 


§100]       SINGLE  AND  SIMULTANEOUS  EQUATIONS         181 

Now  reason  as  follows:  The  actual  root  lies  between  1.32  and 
1.33,  and  the  zero  value  *of  y  corresponds  to  it.  This  zero  is 
200/426  of  the  way  between  the  two  values  of  y;  hence  if  the 
curve  be  nearly  straight  between  x  =  1.32,  and  x  =  1.33, 
the  desired  value  of  x  is  approximately  200/426  of  the  way 
between  1.32  and  1.33  or  it  is  x  =  1.324694.  This  value  is 
probably  correct  to  the  fourth  decimal  place. 

To  find   a   third   approximation  we  build   another  table  of 
values : 

X  \  y 


1.3247 
1.3248 


-  .0000766 
+  .0003499 


0 .  0001        .  0004265     Differences. 

Reasoning  as  before,  we  get  x  =  1.324718  which  is  very  likely 
true  to  the  last  decimal  place. 

The  above  method  is  applicable  to  an  equation  like  exercise 
8  above.  In  fact  it  is  the  only  numerical  method  that  is 
applicable  in  such  cases. 


CHAPTER  VI 

PERMUTATIONS  AND  COMBINATIONS; 
THE  BINOMIAL  THEOREM 

101.  Fudamental  Principle.  //  one  thing  can  be  done  in  n 
different  ways  and  another  thing  can  be  done  in  r  different  ways, 
then  both  things  can  be  done  together,  or  in  succession,  in  n  X  r 
different  ways.  This  simple  theorem  is  fundamental  to  the  work 
of  this  chapter.  To  illustrate,  if  there  be  3  ways  of  going  from 
Madison  to  Chicago  and  7  ways  of  going  from  Chicago  to  New 
York,  then  there  are  21  ways  of  going  from  Madison  to  New 
York. 

To  prove  the  general  theorem,  note  that  if  there  be  only  one 
way  of  doing  the  first  thing,  that  way  could  be  associated  with 
each  of  the  r  ways  of  doing  the  second  thing,  making  r  ways 
of  doing  both.  That  is,  for  each  way  of  doing  the  first,  there  are 
r  ways  of  doing  both  things;  hence,  for  n  ways  of  doing  the  first 
there  are  n  X  r  ways  of  doing  both. 

Illustrations:  A  penny  may  fall  in  2  ways;  a  common 
die  may  fall  in  6  ways;  the  two  may  fall  together  in  12  ways. 

In  a  society,  any  one  of  9  seniors  is  eligible  for  president  and  any 
one  of  14  juniors  is  eligible  for  vice-president.  The  number  of 
tickets  possible  is,  therefore,  9  X  14  or  126. 

I  can  purchase  a  present  at  any  one  of  4  shops.  I  can  give  it 
away  to  any  one  of  7  people.  I  can,  therefore,  purchase  and  give 
it  away  in  any  one  of  28  different  ways. 

A  product  of  two  factors  is  to  be  made  by  selecting  the  first 
factor  from  the  numbers  a,  b,  c,  and  then  selecting  the  second  factor 
from  the  numbers  x,  y,  z,  u,  v.  The  number  of  possible  products 
is,  therefore,  15. 

If  a  first  thing  can  be  done  in  n  different  ways,  a  second  in  r 
different  ways,  and  a  third  in  s  different  ways,  the  three  things 
can  be  done  in  w  Xr  X  s  different  ways.  This  follows  at  once 
from  the  fundamental  principle,  since  we  may  regard  the  first 
two  things  as  constituting  a  single  thing  that  can  be  done  in  nr 

182 


§102]  PERMUTATIONS  AND  COMBINATIONS  183 

ways,  and  then  associate  it  with  the  third,  making  nr  X  s  ways 
of  doing  the  two  things,  consisting  of  the  first  two  and  the  third. 

In  the  same  way,  if  one  thing  can  be  done  in  n  different  ways,  a 
second  in  r  different  ways,  a  third  in  s,  a  fourth  in  t,  etc.,  then  all 
can  be  done  together  in  nXrXsXt.   .   .  different  ways. 

Thus,  n  different  presents  can  be  given  to  x  men  and  a  women 
in  {x  -{-  a)"  different  ways.  For  the  first  of  the  n  presents  can 
be  given  away  in  (x  +  a)  different  ways,  the  second  can  be  given 
away  in  {x  +  a)  different  ways,  and  the  third  in  {x  +  a)  different 
ways  and  so  on.  Hence,  the  number  of  possible  ways  of  giving 
away  the  n  presents  to  {x  +  a)  men  and  women  is: 

{x  -\-  a){x  -\-  a)ix  -}-  a)  .   .    .ton  factors,  or  {x  +  a)« 

102.  Definitions.  Every  distinct  order  in  which  objects 
may  be  placed  in  a  line  or  row  is  called  a  permutation  or  an 
arrangement.  Every  distinct  selection  of  objects  that  can  be 
made,  irrespective  of  the  order  in  which  they  are  placed,  is  called 
a  combination  or  group. 

Thus,  if  we  take  the  letters  a,  b,  c,  two  at  a  time,  there  are  six 
arrangements,  namely:  ab,  ac,  ba,  be,  ca,  cb,  but  there  are  only 
three  groups,  namely:  ab,  ac,  be. 

If  we  take  the  three  letters  all  at  a  time,  there  are  six  arrange- 
ments possible,  namely:  abe,  acb,  bca,  bac,  cab,  eba,  but  there  is 
only  one  group,  namely:  abe. 

Permutations  and  combinations  are  both  results  of  mode  of 
selection.  Permutations  are  selections  made  with  the  understand- 
ing that  two  selections  are  considered  as  different  even  though 
they  differ  in  arrangement  only;  combinations  are  selections  made 
with  the  understanding  that  two  selections  are  not  considered  as 
different,  if  they  differ  in  arrangement  only. 

In  the  following  work,  products  of  the  natural  numbers  like 
1X2X3;     1X2X3X4X5;      etc. 
are  of  frequent  occurrence.     These  products  are  abbreviated  by 
the  symbols  3 !  5 !  and  read  "factorial  three,"  "factorial  five  " 
respectively. 

103.  Formula  for  the  Number  of  Permutations  of  n  Different 
Things  Taken  All  at  a  Time.  We  are  required  to  find  how  many 
possible  ways  there  are  of  arranging  n  different  things  in  a  line. 


184        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§104 

Lay  out  a  row  of  n  blank  spaces,  so  that  each  may  receive  one  of 
these  objects,  thus: 

I     1     I     I    2     I     I    3     i     I    4     I     I     5  .    ,    .   \__n_\ 

In  the  first  space  we  may  place  any  one  of  the  n  objects;  therefore, 
that  space  may  be  occupied  in  n  different  ways.  The  second 
space,  after  one  object  has  been  placed  in  the  first  space,  may  be 
occupied  in  (n  —  1)  different  ways;  hence,  by  the  fundamental 
principle,  the  two  spaces  may  be  occupied  in  n{n  —  1)  different 
ways.  In  like  manner,  the  third  space  may  be  occupied  in  (n  —  2) 
different  ways,  and,  by  the  same  principle,  the  first  three  spaces 
may  be  occupied  in  n{n  —  1)  {n  —  2)  different  ways,  and  so  on. 
The  next  to  the  last  space  can  be  occupied  in  but  two  different 
ways,  since  there  are  but  two  objects  left,  and  the  last  space 
can  be  occupied  in  but  one  way  by  placing  therein  the  last  re- 
maining object.  Hence,  the  total  number  of  different  ways  of 
occupying  the  n  spaces  in  the  row  with  the  n  objects  is  the  product: 

n{n  -  1)  {n  -2)  .    .    .  3  •  2  •  1 
or, 

n! 

If  we  use  the  symbol  P„  to  stand  for  the  number  of  permutations 
of  n  things  taken  all  at  a  time,  then  we  write: 

Pn  =  n!  (1) 

104.  Formula  for  the  Niimber  of  Permutations  of  n  Things 
Taken  r  at  a  Time.  We  are  required  to  find  how  many  possible 
ways  there  are  of  arranging  a  row  consisting  of  r  different  things, 
when  we  may  select  the  r  things  from  a  larger  group  of  n  different 
things. 

For  convenience  in  reasoning,  lay  out  a  row  of  r  blank  spaces, 
so  that  each  of  the  spaces  may  receive  one  of  the  objects,  thus: 

I  1  I  I  2  I  I  3  I  .  .  .  I  r-1  I  I  r  \ 
In  the  first  space  of  the  row,  we  may  place  any  one  of  the  n  objects; 
therefore,  that  space  may  be  occupied  in  n  different  ways.  The 
second  space,  after  one  object  has  been  placed  in  the  first  space, 
may  be  occupied  in  (w  —  1)  different  ways;  hence,  by  the  fun- 
damental principle,  the  two  spaces  may  be  occupied  in  n(n  —  1) 
different  ways.     In  like  manner,  the  third  space  may  be  occupied 


§104]  PERMUTATIONS  AND  COMBINATIONS  185 

in  (n  —  2)  different  ways,  and  hence,  the  first  three  may  be 
occupied  in  n{n  —  1)  {n  —  2)  different  ways,  and  so  on.  The 
last  or  rth  space  can  be  occupied  in  as  many  different  ways  as  there 
are  objects  left.  When  an  object  is  about  to  be  selected  for  the 
rth  space,  there  have  been  used  (r  —  1)  objects  (one  for  each  of 
the  (r  —  1)  spaces  already  occupied).  Since  there  were  n  objects 
to  begin  with,  the  number  of  objects  left  is  n  —  (r  —  1)  or 
n  —  r  +  1,  which  is  the  number  of  different  ways  in  which  the 
last  space  in  the  row  may  be  occupied.     Hence,  the  formula: 

P„..=  n(n  -  l)(n  -  2)    .    .    .    (n  -  r  +  1)  (1) 

in  which  Pn.r  stands  for  the  number  of  permutations  of  n  things 
taken  r  at  a  time. 

The  formula,  by  multiplication  and  division  by  1^  -t,  becomes: 
n(n  -  1)     .    .    .    (n  -  r  +  l)(n  -  r){n  -  r  -  1)   .    .    .    3-2-1 
(n  -r){n  -  r  -  1)   .    .    .      3-2-1 

Pn.=       ^-  (2) 

(n-r)! 

This  formula  is  more  compact  than  the  form  (1)  above,  but  the 
fraction  is  not  in  its  lowest  terms. 

Formula  (1)  is  easily  remembered  by  the  fact  that  there  are 
just  r  factors  beginning  with  n  and  decreasing  by  one.  Thus  we 
have: 

Pio,7  =  10X9X8X7X6X5X4 

Exercises 

1.  How  many  permutations  can  be  made  of  six  things  taken  all  at 
a  time? 

2.  How  many  different  numbers  can  be  made  with  the  five  digits 
1,  2,  3,  4,  5,  using  each  digit  once  and  only  once  to  form  each  number? 

3.  The  number  of  permutations  of  four  things  taken  all  at  a  time 
bears  what  ratio  to  the  number  of  permutations  of  seven  things  taken 
all  at  a  time? 

4.  How  many  arrangements  can  be  made  of  eight  things  taken 
three  at  a  time? 

5.  How  many  arrangements  can  be  made  of  eight  things  taken 
five  at  a  time? 

6.  How  many  four-figure  numbers  can  be  formed  with  the  ten 
digits  0,  1,  2,   .    .    .  9  without  repeating  any  digit  in  any  number? 


186        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§105 

7.  How  many  different  ways  may  the  letters  of  the  word  algebra 
be  written,  using  all  of  the  letters? 

8.  How  many  different  signals  can  be  made  with  seven  different 
flags,  by  hoisting  them  one  above  another  five  at  a  t'me? 

9.  How  many  different  signals  can  be  made  with  seven  different 
flags,  by  hoisting  them  one  above  another  any  number  at  a  time? 

10.  How  many  different  arrangements  can  be  made  of  nine  ball 
players,  supposing  only  two  of  them  can  catch  and  one  pitch? 

105.  Formula  for  the  number  of  combinations  or  groups  of  n 
different  things  taken  r  at  a  time. 

It  is  obvious  that  the  number  of  combinations  or  groups  con- 
sisting of  r  objects  each  that  can  be  selected  from  n  objects,  is 
less  than  the  number  of  permutations  of  the  same  objects  taken 
r  at  a  time,  for  each  combination  or  group  when  selected  can  be 
arranged  in  a  large  number  of  ways.  In  fact,  since  there  are  r 
objects  in  the  group,  each  group  can  be  arranged  in  exactly  r! 
different  ways.  Hence,  for  each  group  of  r  objects,  selected  from 
n  objects,  there  exists  r!  permutations  of  r  objects  each.  There- 
fore, the  number  of  permutations  of  n  things,  taken  r  at  a  time,  is 
r!  times  the  number  of  combinations  of  n  objects  taken  r  at  a 
time.     Calling  the  unknown  number  of  combinations  x,  we  have: 


{n-  r)! 

or,  solving  f or  a; : 

n\ 

X  =  

r\{n  —  r)\ 

This  is  the  number  of  combinations  of  n  objects  taken  r  at  a  time, 

and  may  be  symbolized: 

P      _  n! (1) 

"•'~r!(n-i-)! 

This  fraction  will  always  reduce  to  a  whole  number.     It  may  be 

written  in  the  useful  form: 

n(n-l)(n-2)  .  .  .  (n  -  r  +  1)  (2) 

^''■'  1X2X3         ...  r 

It  is  easily  remembered  in  this  form,  for  it  has  r  factors  in  both 

the  numerator  and  the  denominator.     Thus  for  the  number  of 


§105]  PERMUTATIONS  AND  COMBINATIONS  187 

combinations  of  ten  things  taken  four  at  a  time  we  have  four 
factors  in  the  numerator  and  denominator,  and 

10  X  9  X  8  X  7 


C 


''''     •    1X2X3X4 
Exercises 


1.  How  many  different  products  of  three  each  can  be  made  with  the 
five  numbers  a,  b,  c,  d,  e,  provided  each  combination  of  three  factors 
gives  a  different  product. 

2.  How  many  products  can  be  made  from  twelve  different  num- 
bers, by  taking  eight  numbers  to  form  each  product? 

3.  How  many  products  can  be  made  from  twelve  different  num- 
bers, by  taking  four  numbers  to  form  each  product? 

4.  How  many  different  hands  of  thirteen  cards  each  can  be  held 
at  a  game  of  whist? 

5.  In  how  many  ways  can  seven  people  sit  at  a  round  table  ? 

6.  In  how  many  ways  can  a  child  be  named,  supposing  that  there 
are  400  different  Christian  names,  without  giving  it  more  than  three 
names? 

7.  In  how  many  ways  can  a  committee  of  three  be  appointed 
from  six  Germans,  four  Frenchmen,  and  seven  Americans  provided 
each  nationality  is  represented? 

8.  There  are  five  straight  lines  in  a  plane,  no  two  of  which  are 
parallel;  how  many  intersections  are  there? 

9.  There  are  five  points  in  a  plane,  no  three  of  which  are  collinear ; 
how  many  lines  result  from  joining  each  point  to  every  other  point? 

10.  In  a  plane  there  are  n  straight  lines,  no  two  of  which  are  parallel; 
how  many  intersections  are  there? 

11.  In  a  plane  there  are  n  points,  no  three  of  which  are  collinear; 
how  many  straight  lines  do  they  determine? 

12.  In  a  plane  there  are  n  points,  no  three  of  which  are  collinear, 
except  r,  which  are  all  in  the  same  straight  line;  find  the  number  of 
straight  lines  whch  result  from  joining  them. 

13.  A  Yale  lock  contains  five  tumblers  (cut  pins),  each  capable  of 
being  placed  in  ten  distinct  positions.  At  a  certain  arrangement  of  the 
tumblers,  the  lock  is  open.  How  many  locks  of  this  kind  can  be  made 
so  that  no  two  shall  have  the  same  key? 

14.  In  how  many  ways  can  seven  beads  of  different  colors  be  strung 
so  as  to  form  a  bracelet? 


188        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§106 

15.  How  many  different  sums  of  money  can  be  formed  from  a  dime, 
a  quarter,  a  half  dollar,  a  dollar,  a  quarter  eagle,  a  half  eagle,  and  an 
eagle? 

106.*  The  Arithmetical  Triangle.  In  deriving  by  actual  mul- 
tiplication, as  below,  any  power  of  a  binomial  x  -{-  a  from  the 
preceding  power,  it  is  easy  to  see  that  any  coefficient  in  the  new 
power  is  the  sum  of  the  coefficient  of  the  corresponding  term  in  the 
multiplicand  and  the  coefficient  preceding  it  in  the  multiplicand. 
Thus: 

x^  +  3ax2  +  Sa^x  +  a^ 
X  +  a 


x^  +  3ax3  +  Sa'^x^  +    a^x 

ax^  -\-  Sa'^x'^  +  3a%  +  a'* 
x*  +  4ax3  +  6aH^  +  4:a^x  +  a* 
or,  erasing  coefficients,  we  have: 

1+3+3+1 

1  +  1 

1+3+3+1 
1+3+3+1 


1+4+6+4+1 

from  which  the  law  of  formation  of  the  coefficients  1,  4,  6,  .  .  . 
is  evident.  Hence,  writing  down  the  coefficients  of  the  powers 
of  a;  +  a  in  order,  we  have : 

Powers  Coefficients 


1    2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

0    ] 

1    ] 

L     1 

2    ] 

L     2 

1 

3    ] 

L     3 

3 

1 

4    ] 

L     4 

6 

4 

1 

5    ] 

L     5 

10 

10 

5 

1 

6    ] 

L     6 

15 

20 

15 

6 

1 

7    ] 

L     7 

21 

35 

35 

21 

7 

1 

8    ] 

L     8 

28 

56 

70 

56 

28 

8 

1 

9    ] 

L     9 

36 

84 

126 

126 

84 

36 

9 

1 

10 

L    10 

45 

120 

210 

252 

210 

120 

45 

10 

1 

[§107  PERMUTATIONS  AND  COMBINATIONS  189 

In  this  triangle,  each  number  is  the  sum  of  the  number  above  it 
and  the  number  to  the  left  of  the  latter.  Thus  84  in  the  9th  line 
equals  56  +  28,  etc.  The  triangle  of  numbers  was  used  previous 
to  the  time  of  Isaac  Newton  for  finding  the  coefficients  of  any  de- 
sired power  of  a  binomial.  At  that  time  it  was  little  suspected 
that  the  coefficients  of  any  power  could  be  made  without  first 
obtaining  the  ocefficients  of  the  preceding  power.  Isaac  Newton, 
while  an  undergraduate  at  Cambridge,  showed  that  the  coefficients 
of  any  power  could  be  found  without  knowing  the  coefficients  of 
the  preceding  power;  in  fact,  he  showed  that  the  coefficients  of 
any  power  n  of  a  binomial  were  functions  of  the  exponent  n. 

The  above  triangle  of  numbers  is  known  as  the  arithmetical 
triangle  or  as  Pascal's  triangle. 

107.  Distributive  Law  of  Multiplication.  The  demonstration 
of  the  binomial  theorem  may  be  based  upon  the  following  law  of 
multiplication:  The  product  of  any  number  of  polynomials  is 
the  aggregate  of  all  the  possible  partial  products  which  can  be  made 
by  taking  one  term  and  only  one  from  each  of  the  polynomials. 
This  statement  is  merely  a  definition  of  what  is  meant  by  the 
product  of  two  or  more  polynomials.     (See  appendix.)     Thus: 

{x  H-  a){y  +  b){z  +  c)  = 
xyz  +  ayz  +  bxz  +  cxy  +  abz  +  bcx  +  cay  +  abc 
Each  of  the  eight  partial  products  contains  a  letter  from  each 
parenthesis,  and  never  two  from  the  same  parenthesis.  The 
number  of  terms  is  the  number  of  different  ways  in  which  a  letter 
can  be  selected  from  each  of  the  three  parentheses.  In  the  present 
case  this  is,  by  §101,  2X2X2  =  8. 

108.  Binomial  Formula.  It  is  required  to  write  out  the  value 
of  {x  -\-  a)",  where  x  and  a  stand  for  any  two  numbers  and  n  is  a 
positive  integer.  That  is,  we  must  consider  the  product  of  the  n 
parentheses: 

{x  -\-  a){x  -\-  a){x  -\-  a)    .    .    .    (x  +  a) 
by  the  distributive  law  stated  above. 

First.  Take  an  x  from  each  of  the  parentheses  to  form  one  of 
the  partial  products.     This  gives  the  term  x»  of  the  product. 

Second.  Take  an  a  from  the  first  parenthesis  with  an  x  from 
each  of    the  other   (n  —  1)    parentheses.     This  gives  ax«~i  as 


190        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§108 

another  partial  product.  But  if  we  take  a  from  the  second  paren- 
thesis and  an  x  from  each  of  the  other  (n  —  1)  parentheses,  we  get 
ax""!  as  another  partial  product.  Likewise  by  taking  a  from  any 
of  the  parentheses  and  an  a:  from  each  of  the  other  (n  —  l)  paren- 
theses, we  shall  obtain  ax«-^  as  a  partial  product.  Hence,  the 
final  product  contains  n  terms  like  ax""-^,  or  wax"~^  is  a  part 
of  the  product. 

Third.  We  may  obtain  a  partial  product  like  a^x'*"^  by  taking 
an  a  from  any  two  of  the  parentheses,  together  with  the  x's  from 
each  of  the  other  (n  —  2)  parentheses.  Hence,  there  are  as  many 
partial  products  like  a^^n-s  as  there  are  ways  of  selecting  two  a's 
from  n  parentheses;  that  is,  as  many  ways  as  there  are  groups  or 
combinations  of  n  things  taken  two  at  a  time,  or: 

n{n  —  1) 
V2 

Hence,  — r^ —  a^'x"-^  is  another  part  of  the  product. 

Fourth.  We  may  obtain  a  partial  product  like  aH''-^  by  taking 
an  a  from  any  three  of  the  parentheses  together  with  the  x's  from 
each  of  the  other  (n  —  3)  parentheses.  Hence,  there  are  as  many 
partial  products  like  a^x""'^  as  there  are  ways  of  selecting  three  a'S 
from  n  parentheses,  that  is,  as  many  ways  as  there  are  combin^- 

YifYi 1  ^  (n  2  \ 

tions  of  n  things  taken  three  at  a  time,  or f^oTq ■' 

Hence,  V2^ a^x**-^  is  another  part  of  the  product. 

In  general,  we  may  obtain  a  partial  product  like  a'"a:«~'"  (where  r 
is  an  integer  <  n)  by  taking  an  a  from  any  r  of  the  parentheses 
together  with  the  x's  from  each  of  the  other  {n  —  r)  parentheses. 
Hence,  there  are  as  many  partial  products  like  a'"x'»~'"  as  there  are 
ways  of  selecting  r  a's  from  n  parentheses;  that  is,  as  many  ways 
as  there  are  combinations  of  n  things  taken  r  at  a  time,  or 

Hence,   -y-. — '^ — t  ,  a''x''~''  stands   for  any  term 


r\  {n  —  r)\  '    r!  {n  —  r)\ 

in  general  in  the  product  (re  +  0)". 

Finally,  we  may  obtain  one  partial  product  like  a"  by  taking  an 
a  from  each  of  the  parentheses.  Hence,  a"  is  the  last  term  in  the 
product. 


§109]  PERMUTATIONS  AND  COMBINATIONS  191 

Thus  we  have  shown  that: 

(x  +  a)>>  =  x»  4-  nax^-^  +     \Z  -  a^x^^-^  +    .    .    . 

n! 

^  r!(n-r)!"      t-   .    •    •   -h  a 

This  is  the  binomial  formula  of  Isaac  Newton.  The  right  side  is 
called  the  expansion  or  development  of  the  power  of  the  binomial. 

It  is  obvious  that  the  expansion  of  (x  —  a)"  will  differ  from  the 
above  only  in  the  signs  of  the  alternate  terms  containing  the  odd 
powers  of  a,  which,  of  course,  will  have  the  negative  sign. 

109.  Binomial  Theorem.  The  binomial  expansion  is  a  series, 
that  is,  each  term  may  be  derived  from  the  preceding  term  by  a 
definite  law.  This  law  is  made  up  of  two  parts  which  may  be 
stated  as  follows: 

(1)  Law  of  Exponents.  In  any  power  of  a  binomial,  x  -\-  a,  the 
exponent  of  x  commences  in  the  first  term  with  the  exponent  of  the 
required  power,  and  in  the  following  terms  continually  decreases  by 
unity.  The  exponent  of  a  commences  with  1  in  the  second  term  and 
continually  increases  by  unity. 

(2)  Law  of  Coefficients.  The  coefficient  in  the  first  term  is  1, 
that  in  the  second  term  is  the  exponent  of  the  power;  and  if  the 
coefficient  in  any  term  be  multiplied  by  the  exponent  of  x  in  that 
term  and  divided  by  the  exponent  of  a,  increased  by  1,  it  will  give  the 
coefficient  in  the  succeeding  term. 

Exercises 

1.  Expand  {u  +  Sy)^.     Here  x  =  u  and  a  =  3y.     By  the  formula 
we  get: 

u^  +  5u'(Sy)  +  lOu^Sy)^  +  lOu^Sy)^  +  5u{Syy+  {3y)' 
Performing  the  indicated  operations,  we  obtain : 

u^  +  15u*y  +  90^3^2  +  270u^y^+  405w^*  +  24Sy^ 
Expand  each  of  the  following  by  the  binomial  formula: 

2.  (r2  -  2)*.  8.  (1/2  +  x)K 

3.  (36  -  1/2)6.  9.  (52  _  c2)6. 

4.  (c  +  x)«.  10.  (3a  +  l/2)«. 
6.  (2x2  _  a.)6  11.  (5(i  _  32^)6. 

6.  (1  -o)8.  12.  (3x3-  1)4. 

7.  (-X  +2ay.  13.  iVa  +  x)«. 


192        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§110 

14.  {x^^   +  x'^^  )6.  17.  (a  +  [x  +  y])K 

16.  (a-2_-  b'"^  y^  18.  (a  +  6  -  i/)3. 

16.  (Vab  -  Vab)'.  19.  {x^  +  2ax  +  a^)^ 

110.  Binomial  Theorem  for  Fractional  and  Negative  Exponents. 
It  is  proved  in  the  Calculus  that: 

/I    1     \         1    ,  ,    w(n  —  1)    „    ,   n  (n  —  1)  (n  —  2)    ,    . 

(1  ±  x)-  =  1  ±  nx  -\ ^,        x*  +  gP -x'  +  .  .  . 

is  true  for  fractional  and  for  negative  values  of  n,  provided  x  is 
less  than  1  in  absolute  value.  The  number  of  terms  in  the  expan- 
sion is  not  finite,  but  is  unlimited,  and  the  series  or  expansion 
converges  or  approaches  a  definite  hmit  as  the  number  of  terms  of 
the  expansion  is  increased  without  limit,  provided  \x\  <  1. 
By  the  above  formula,  we  have: 

VYT-.=  1^(1/2).  +  '^^^-^.^ 

(1/2)  (1/2  -  1)  (1/2  -  2)   3_^ 
-r  3j  X   -h  .    .    . 


=  1  +  (1/2 )a;  -  (1/8)^2  +  (l/16)a:3  -  (5/128)^4 
If 

a;  =  1/2 
this  becomes: 

VSI2  =  1  +  1/4  -  1/32  +  1/128  -  5/2048  +  .    .    . 
Therefore,  using  five  terms  of  the  expression: 

The  square  root,  correct  to  four  figures,  is  really  1.2247.  Thus  the 
error  in  this  case  is  less  than  one-tenth  of  1  percent  if  only  five 
terms  of  the  series  be  used.  The  degree  of  accuracy  in  each  case 
is  dependent  both  upon  the  value  of  n  and  upon  the  value  of  x. 
Obviously,  for  a  given  value  of  n,  the  series  converges  for  small 
values  of  x  more  rapidly  than  for  larger  values. 

As  another  example,  suppose  it  is  required  to  expand  (1  —  x)-K 
By  the  binomial  theorem: 


§1111  PERMUTATIONS  AND  COMBINATIONS  193 

(1  -  X)-'  =  1  +  (-  1)(  -  X)  +  =-^^^-^^-  (  -  ^V 

=  1  +  a;  +  x2  +  ^3  +   .    .    . 
If  five  terms  of  the  series  be  used,  the  error  is  1  /16  f or  x  =  1  /2, 
or  about  3  percent. 

111.  Approximate  Formulas.     If  x  be  very  small,  the  expansion 
of: 

{l  +  x)n=l  +  nx  +  ^^—h^+    .    .    . 

Ji ! 

is  approximately: 

(l  +  a:)"^l  +  nx  (1) 

since  x"^,  x^  and  all  higher  powers  of  x  are  much  smaller  than  x. 
Thus,  using  the  symbol  =j=  to  express  ^'approximately  equals,"  we 
have,  for  example: 

(1.01)3  =T=  1.03 
for  (1  +  1/100)3  ^  1  +  3/100 

The  true  value  of  (l.Ol)^  is  1.030301,  so  that  the  approximation  is 
very  good. 
Likewise: 

(1  _  x)n  =  1  -  nx  (2) 

if  X  be  small. 

If  X,  y,  and  z  be  small  compared  with  unity,  the  following 
approximate  formulas  hold: 

(H-x)(l  +  y)  ^1  +  x  +  y  (3) 

(l  +  x)/(l  +  y)  =Fl  +  x-y  (4) 

(l  +  x)(l  +  y)(l  +  z)  -1  +  x  +  y  +  z  (5) 

The  approximation  formulas  are  proved  as  follows: 

(1  +  x)(l  +  ?/)  =  l-\-x-\-y-\-xy^\-\-x-\-y,iovxy  is  small 

compared  to  x  and  y. 

?nt~'i  =  1  +  x  —  2/  +  ».-—-—?-  =  I  j^  X  —  y,iox  the  fraction  is 

small  compared  to  x  and  y. 

(1  +  x)  (1  +  2/)  (1  +  2)  -  (1  +  a;  +  ?/)  (1  +  2)  -  1  +  a;  +  y  +  2 

13 


194        ELEMENTARY  MATHEMATICAL  ANALYSIS       l§112 


112.  *  The  Progressive  Mean.  In  using  scientific  data  it  is  often 
desirable  to  determine  the  so-called  progressive  mean  of  a  highly- 
fluctuating  magnitude.  Thus  if  we  wish  to  determine  whether 
or  not  the  rainfall  at  New  York  has  on  the  average  been  increasing 
or  decreasing  in  the  last  100  years,  we  form  an  average  for  each 
successive  group  of  five  or  six  or  seven  or  other  convenient  number 
of  years,  and  tabulate  and  compare  these  averages.  In  finding 
these    averages,  however,    the   various    years    are  weighted  as 


'^n               "^               §1-1 

ri 

«             ?        r 

-, 

s            i        ^S 

'\       r 

1         /  I      ^\ 

on    :^                f                (X        7                       \ 

/    J   \     1   \n 

""  _^ 1 r.  a_/ '"»-. 

1        ^,7_?r               T 

ixi c/---  — 

,i    ■^-" 

10 

5 

„  11 11  Pii  ii  ill  II  iii»  11 

illliiililiiiiliiiiliii 

Fig.    84. — Annual,     Mean,    and   Progressive    Mean    Rainfall     (by    5-yr- 
Periods)  at  Dodge,  Kansas. 

follows:  If  the  numbers  whose  progressive  means  are  desired  be 
di,  a2,  as,  tti,  .  .  .  ,  then  the  progressive  mean  corresponding 
to  ttio  would  be,  for  five-year  intervals, 

m  =  (as  +  4a9  +  6aio  +  4aii  +  012)  /16 
and  for  seven-year  intervals, 

m  =  (a?  +  6^8  +  15a9  +  20aio  +  15an  +  6ai2  +  a]3)/64 
In  these  expressions  the  coefficients  are  the  binomial  coefficients 
and  the  divisors  are  the  sum  of  the  coefficients.     See  Fig.  84. 


112]  PERMUTATIONS  AND  COMBINATIONS  195 

Exercises 
1.  Explain  the  following  approximate  formulas,  in  which  \x\  <  1 


Vl  -fa;  ?  1  H-(l/2)x 

Vl  -  X  =7=  1  -  (l/2)a; 

(1  -f^r^-  1  -  (l/2)x 

Vl  +x=F  1  +  (l/3)x 

Vl    -X=F    1    -  (1/S)X 

(1  +  x)-^^  ^  1  -  (2/3)  X 
(1  +a:2)>^^  1  +  (l/3)a;2. 

2.  Compute  the  numerical  value  of : 

(1.03)'-'^  (1.05)'^^ 

(1.02)  (1.03)  1.02/1.03 

(1.01)(1.02)/(1.03)(1.04). 

3.  The  formula  for  the  period  of  a  simple  pendulum  is : 

T  =WTfg 
For  the  value  of  gravity  at  New  York,  this  reduces  to 

6.253 
in  which  I,  the  length  of  the  pendulum,  is  measured  in  inches.     This 
pendulum  beats  seconds  when 

I  =  (6.253)2  or  39.10  inches. 
What  is  the  period  of  the  pendulum  if  I  be  lengthened  to  39.13  inches? 
Hint: 

^        6.253 

^    "  "6.253"    "  6.253  ^^  +  ^^^ 

=  6^3(1+^/20 

Take  I  =  39.10,  and  h  =  0.03 
Then: 

r  =  1  +  0.03/78.20 
=  1.00038. 

A  day  contains  86,400  seconds.  The  change  of  length  would,  there- 
fore, cause  a  loss  of  32.8  seconds  per  day,  if  the  pendulum  were  attached 
to  a  clock. 


196        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§H2 

4.  On  the  ocean  how  far  can  one  see  at  an  elevation  of  h  feet  above 
its  surface? 

Call  the  radius  of  the  earth  a(=  3960  miles),  and  the  distance 
one  can  see  d,  which  is  along  a  tangent  from  the  point  of  observation 

to  the  sphere.     Since  h  is  in  feet,  and  a  +  Horo'  ^'  ^"^^  ^  ^^®  ^^®  sides 
of  a  right  triangle,  we  have  (a  +  /i/5280)2  =  d^  +  a^ 
or:  a2(l  +  /i/5280a)2  =  d^  +  a\ 


Fig.  85. — Graphical  Representation  of  the  Values  of  the  Binomial 
Coefficients  in  the  999th  power  of  a  Binomial.  The  middle  coefficients  are 
taken  equal  to  5,  for  convenience,  and  the  others  are  expressed  to  that 
scale  also. 

Expanding  by  the  approximate  formula : 

a2(l  +  2/i/5280a)  =  d^  +  a^ 


or: 


or: 


d^  =  2a/t/5280 

=  2  X  3960/i/5280 

=  (3/2)/i 


d  =  V(3/2)/i 
where  d  is  expressed  in  miles  arid  h  is  in  feet.     See  §66,  exercise  13. 


§113]  PERMUTATIONS  AND  COMBINATIONS  197 

6.  How  much  is  the  area  of  a  circle  altered  if  its  radius  of  100  cm- 
be  changed  to  101  cm.? 

6.  How  much  is  the  volume  of  a  sphere,  iira^f  altered  if  the  radius 
be  changed  from  100  cm.  to  101  cm.? 

7.  If  the  formula  for  the  horse  power  of  a  ship  is  I.H.P.  =    ortr> 

where  S  is  speed  in  knots  and  D  is  displacements  in  tons,  what  in- 
crease in  horse  power  is  required  in  order  to  increase  the  speed  from 
fifteen  to  sixteen  knots,  the  tonnage  remaining  constant  at  5000? 
What  increase  in  horse  power  is  required  to  maintain  the  same  speed 
if  the  load  or  tonnage  be  increased  from  5000  to  5500? 

113.  *  Graphical  Representation  of  the  Coefficients  of  any  Power 
of  a  Binomial.  If  we  erect  ordinates  at  equal  intervals  on  the 
X-axis  proportional  to  the  coefficients  of  any  power  of  a  binomial, 
we  find  that  a  curve  is  approximated,  which  becomes  very  striking 
as  the  exponent  is  taken  larger  and  larger.  In  Fig.  85,  the  ordi- 
nates are  proportional  to  the  coefficients  of  the  999th  power  of 
{x  -{-  a).     The  drawing  is  due  to  Quetelet. 

The  limit  of  the  broken  line  at  the  top  of  the  ordinates  in  Fig.  85 
is,  as  n  is  increased  indefinitely,  a  bell-shaped  curve,  known  as 
the  probability  curve;  its  equation  is  of  the  form  y  =  ae"^^^,  as 
is  shown  in  treatises  on  the  Theory  of  Probability. 


CHAPTER  VII 
PROGRESSIONS 

114.  An  Arithmetical  Progression  or  an  Arithmetical  Series, 

is  any  succession  of  terms  such  that  each  term  differs  from  that 
immediately  preceding  by  a  fixed  number  called  the  common 
difference.     The  following  are  arithmetical  progressions: 

(1)  1,  2,  3,  4,  5. 

(2)  4,  6,  8,  10,  12. 

(3)  32,  27,  22,  17,  12. 

(4)  2i,  31,  5,  6i,  7|. 

(5)  (u  -  v),  u,  (u  +  v). 

(6)  a,  a  +  d,  a  -{-  2d,  a  +  Sd,   .    .    . 

The  first  and  last  terms  are  called  the  extremes,  and  the  other 
terms  are  called  the  means. 

Where  there  are  but  three  numbers  in  the  series,  the  middle 
number  is  called  the  arithmetical  mean  of  the  other  two.  To  find 
the  arithmetical  mean  of  the  two  numbers  a  and  b,  proceed  as 
follows: 

Let  A  stand  for  the  required  mean;  then,  by  definition: 
A  —  a  =  b  —  A 
whence: 

A  =  (a  +  6)  /2 
Thus,  the  arithmetical  mean  of  12  and  18  is  15,  for  12,  15,  18  is  an 
arithmetical  progression  of  common  difference  3. 

By  the  arithmetical  mean  or  arithmetical  average  of  several 
numbers  is  meant  the  result  of  dividing  the  sum  of  the  numbers  by 
the  number  of  the  numbers.  It  is,  therefore,  such  a  number  that 
if  all  numbers  of  the  set  were  equal  to  the  arithmetical  mean,  the 
sum  of  the  set  would  be  the  same. 

The  general  arithmetical  progression  of  n  terms  is  expressed  by: 
Number  of 

term:  12  3  4  ...  n 

Progression:  a,  {a  -\-  d) ,  {a  +  2d),  (a  +  Sd),   .    .    .   (a  +  [n  —  l]d) 

198 


§115]  PROGRESSIONS  199 

Here  a  and  d  may  be  any  algebraic  numbers  whatsoever,  integral 
or  fractional,  rational  or  irrational,  positive  or  negative,  but  n 
must  be  a  positive  integer.  If  the  common  difference  be  negative, 
the  progression  is  said  to  be  a  decreasing  progression;  otherwise, 
an  increasing  progression. 

From  the  general  progression  written  above,  we  see  that  a  for- 
mula for  deriving  the  nth  term  of  any  progression  may  be  written: 

1  =  a  +  (n  -  l)d  (1) 

in  which  I  stands  for  the  nth  term. 

115.  The  Sum  of  n  Terms.  If  s  stands  for  the  sum  of  n  terms 
of  an  arithmetical  progression,  and  if  the  sum  of  the  terms  be 
written  first  in  natural  order,  and  again  in  reverse  order,  we  have: 

s  =  a  +  (a  +  d)  +  (a  +  2ci)  +  •  •  •  +  (a -\- [n  -  l]d)  (1) 
s=  I  -^{l  -d)-^{l  -2d)+   .    .    .   +  (Z  -  [n  -  m         (2) 

Adding  (1)  and  (2),  term  by  term,  noting  that  the  positive  and 
negative  common  differences  nuUify  one  another,  we  obtain: 

2s  =  (a  +  0  +  («  +  0  +  («  +  0  +   .    .    .   +  (a  +  0      (3) 

or,  since  the  number  of  terms  in  the  original  progression  is  n,  we 
may  write: 

2s  =  n(a  +  I) 
or:  s  =  n(a+l)/2  (4) 

In  the  above  expression,  (a  +  1)12  is  the  average  of  the  first  and 
nth  terms.  The  formula  (4)  states,  therefore,  that  the  sum  equals 
the  number  of  the  terms  multiphed  by  the  average  of  the  first  and 
last. 

116.  An  arithmetical  progression  is  a  very  simple  particular  in- 
stance of  a  much  more  general  class  of  expressions  known  in  mathe- 
matics as  series.  A  series  is  any  sequence  of  terms  formed  accord- 
ing to  some  law,  such  as: 

(x+  1)  +  (:r  +  2)2-f  (.'c  +  3)^   .    .    . 

X -{- Sx^  -{-  5x' +   .    .    . 

cos  X  +  COS  2x  +  cos  3x  4-   .    .    . 

It  is  only  in  a  very  limited  number  of  cases  that  a  short  expression 

can  be  found  for  the  sum  of  n  terms  of  a  series.     An  arithmetical 

progression  is  one  of  these  exceptions. 


200        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§117 


K 

h/ 


/ 


r 


r 


t 


f 


117.  The  formulas  (1)  and  (4)  above  are  illustrated  graphically 
by  Fig.  86.     Ordinates  proportional  to  the  terms  of  a  progression 

are  laid  off  at  equal  intervals  on  the 
line  OX.  The  ends  of  these  lines,  be- 
cause of  the  equal  increments  in  the 
terms  of  the  series,  lie  on  the  straight 
line  MN.  By  reversing  terms  and 
adding,  the  sums  lie  within  the  rec- 
tangle OK  whose  altitude  is  (a  +  Z). 

The  sum  of  an  arithmetical  pro- 
gression is  readily  constructed.  On 
OY,  lay  off  the  unit  of  measure  01; 
and,  to  the  same  scale,  n.  On  OX, 
lay  off  (a  +  /)•  From  2  on  OF  draw 
a  line  to  {a  +  I)  on  OX.  From  n  on 
0  Y  draw  a  parallel  to  the  latter,  cut- 
ting OX  in  s,  the  required  sum.  This 
construction  has  little  value,  except 
that  it  illustrates  that  s,  for  all 
values  of  a  and  d,  increases  indefi- 
nitely in  absolute  value  as  n  increases 
without  limit,  or,  using  the  equiva- 
lent terms  already  explained,  that 
s  becomes  infinite  as  n  becomes  in- 
finite. 

118.  Formula  (1),  §114,  enables  us  to  obtain  the  value  of  any 
one  of  the  numbers,  I,  a,  n,  d,  when  three  are  given.     Thus: 

(1)  Find  the  100th  term  of: 

3  -h  8  +  13  +    .    .    . 
Here:  a  =  S,  d  =  5,  n  =  100 

therefore,  Z  =  3  +  99  X  5  =  498 

(2)  Find  the  number  of  terms  in  the  progressiqn: 

5  +  7  +  9-f-...+39 
Here:  a  =  5,  cZ  =  2,  Z  =  39 

whence:  39  =  5  +  (n  -  1)2 

Solving  for  n:  n  =  18 


L 


12    3    4    5    6    7 


Fig.  86. — Graphical  Determi- 
nation of  the  Sum  of  an  A.P 


§118]  PROGRESSIONS  201 

(3)  Find  the  common  difference  in  a  progression  of  fifteen  terms 
in  which  the  extremes  are  1/2  and  A2^: 
Here:  a  =  1/2,Z  =  42^71  =  15 

whence:  42j  =  1/2  +  (15  -  l)d 

Solving:  d  =  S 

Formula  (4) ,  §115,  enables  us  to  find  the  value  of  any  one  of  the 
numbers  s,  n,  a,  I,  when  the  values  of  the  other  three  are  given. 
Thus: 

(5)  Find  the  number  of  terms  in  an  arithmetical  progression  in 
which  the  first  term  is  4,  the  last  term  22,  and  the  sum  91. 
Here:  a    =  4,  Z  =  22,  s  =  91 

whence:  91  =  w(4  +  22)  /2 

solving  for  n:  n  =  7 

The  two  formulas,  (1)  §114  and  (4)  §115,  contain  five  letters; 
hence,  if  any  two  of  them  stand  for  unknown  numbers,  and  the 
values  of  the  others  are  given,  the  values  of  the  two  unknown 
numbers  can  be  found  by  the  solution  of  a  system  of  two  equa- 
tions.    Thus: 

(6)  Find  the  number  of  terms  in  a  progression  whose  sum  is 
1095,  if  the  first  term  is  38  and  the  difference  is  5. 

Here:  s  =  1095,  a  =  38,  and  d  =  5 

whence:  Z  =  38  +  (n  -  1)5  (1) 

1095  =  n(38  +  I)  /2  (2) 

From(l):  Z  =  33  +  5n  (3) 

From  (2) :  2190  =  38n  +  nl  (4) 

Substituting  the  value  of  I  from  (3)  in  (4),  we  get: 

2190  =  7ln  +  5n2  (5) 

Solving  this  quadratic,  we  find: 

n  =  15,  or  -  29.2 
The  second  result  is  inadmissible,  since  the  number  of  terms 
cannot  be  either  negative  or  fractional. 

Exercises 

Solve  each  of  the  following: 

1.  Given,  a  =  7,  d  =  4,  n  =  15;  find  I  and  s. 

2.  Given,  a  =  17,  Z  =  350,  d  =  9;  find  n  and  s. 


202        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§119 

3.  Given,  a  =  3,  n  =  50,  s  =  3825;  find  I  and  d. 

4.  Given,  s  =  4784,  o  =  41,  d  =  2;  find  I  and  n. 

5.  Given,  s  =  1008,  d  =  4,  I  =  8S;  find  a  and  n. 

6.  Find  the  sum  of  the  first  n  even  numbers. 

7.  Find  the  sum  of  the  first  n  odd  numbers. 

8.  Insert  nine  arithmetical  means  between  —  7/8  and  +  7/8. 

9.  Sum  (a  +  6)2  +  (a^  +  b^)  +  (a  -  6)2  to  n  terms. 

10.  Find  the  sum  of  the  first  fifty  multiples  of  7. 

11.  Find  the  amount  of  $1.00  at  simple  interest  at  5  percent  for 
1912  years. 

12.  How  long  must  $1.00  accumulate  at  3|  percent  simple  inter- 
est until  the  total  amounts  to  $100? 

13.  How  many  terms  of  the  progression  9  +  13  +  17  + 

must  be  taken  in  order  that  the  sum  may  equal  624?     How  many 
terms  must  be  taken  in  order  that  the  sum  may  exceed  750? 

14.  Show  that  the  only  right  triangle  whose  sides  are  in  arithmetical 
progression  is  the  triangle  of  sides  3,  4,  5,  or  a  triangle  with  sides  pro- 
portional to  these  numbers. 

119.  Geometrical  Progression.  A  geometrical  progression 
is  a  series  of  terms  such  that  each  term  is  the  product  of  the 
preceding  term  by  a  fixed  factor  called  the  ratio.  The  following 
are  examples: 

(1)  3,  6,  12,  24,  48. 

(2)  100,  -50,  25,  -12i 

(3)  1/2,   1/4,  1/8,  1/16,   1/32. 

(4)  a,  ar,  ar"^,  ar^,  ar^  .    .    . 

The  geometrical  mean  of  two  numbers,  a  and  b,  is  found  as 
follows:  Let  G  stand  for  the  required  mean.  Then,  by  the 
definition  of  a  geometrical  progression: 

G/a  =b/G 
whence: 

or:  G^  =  ab  ^ 

G  =  \/'ab 

Thus,  4  is  the  geometrical  mean  of  2  and  8.  The  arithmetical 
mean  of  2  and  8  is  5.  The  geometrical  mean  of  n  positive  num- 
bers is  the  value  of  the  nth  root  of  their  product.  Thus  the  geo- 
metrical mean  of:  

8,  9  and  24  is  12  =   -^  8  X  9  X  24 


§120]  PROGRESSIONS  203 

120.  The  nth  Term  and  the  Sum  of  n  Terms.  If  a  represents 
the  first  term  and  r  the  ratio  of  any  geometrical  progression,  the 
progression  may  be  written: 

Number  of  term:    1     2     3       4     .    .    .  n  —  1    n 
Progression:  a,  ar,  ar"^,  ar^,   .    .    .  ar'^'^,  ar^"-^ 

Therefore,  representing  the  nth  term  by  Z,  we  obtain  the  simple 
formula: 

1  =  ar»-i  (1) 

Representing  by  s  the  sum  of  n  terms  of  any  geometrical  pro- 
gression, we  have: 

s  =  a  -\-  ar  -\-  ar^  +    .    .    .    +  ar""'^  +  ar""^ 
Factoring  the  right  member: 

s  =  a(l  +  r  +  r2  4-    .    .    .    +  r"-2  +  r"-i) 

But,  by  a  fundamental  theorem  in  factoring,  ^  the  expression  in  the 
parenthesis  is  the  quotient  of  1  —  r"  by  1  —  r.     Hence: 

s  =  a(l  -r")/(l-r)  (2) 

Another  form  is  obtained  by  introducing  I  by  the  substitution: 

ar"~i  =  I 

s  =  (a-rl)/(l-r)  (3) 

121.  Formula  (1),  or  (2),  enables  one  to  find  any  one  of  the  four 
numbers  involved  in  the  equations  when  three  are  given.  The 
two  formulas  (1)  and  (2)  considered  as  simultaneous  equations 
enable  one  to  find  any  two  of  the  five  numbers  a,  r,  n,  I,  s,  when  the 
other  three  are  given.  But  if  r  be  one  of  the  unknown  numbers,  the 
equations  of  the  system  may  be  of  a  high  degree,  and  beyond  the 
range  of  Chapter  VII,  unless  solved  by  graphical  means.  If  n  be 
an  unknown  number,  an  equation  of  a  new  type  is  introduced, 
namely,  one  with  the  unknown  number  appearing  as  an  exponent. 
Equations  of  this  type,  known  as  exponential  equations,  will  be 
treated  in  the  chapter  on  logarithms.  The  following  examples 
illustrate  cases  in  which  the  resulting  single  and  simultaneous 
equations  are  readily  solved. 

(1)  Insert  three  geometrical  means  between  31  and  496. 
Here: 

a  =  31,     I  =  496,  and  n  =  5 

1  See  Appendix. 


204        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§121 

whence: 

496  =  31  X  r* 
or: 

r*  =  16 
therefore: 

r  =  ±  2 

consequently,  the  required  means  are  either  62,   124,  and  248, 
or  -  62,  +  124,  and  -  248. 

(2)  Find  the  sum  of  a  geometrical  progression  of  five  terms, 
the  extremes  being  8  and  10,368. 

Here: 

a  =  8,     I  =  10,368,  and  n  =  5 
whence: 

10,368  =  8r4  (1) 

s  =  (10,368r  -  8)  /(r  -  1)  (2) 

From  the  first, 

r  =  6 
whence,  from  the  second, 

s  =  12,440 

(3)  Find  the  extremes  of  a  geometrical  progression  whose  sum 
is  635,  if  the  ratio  be  2  and  the  number  of  terms  be  7. 

Here: 

s  =  635,  r  =  2,  and  n  =  7 
whence: 

I  =  a-2«  (1) 

635  =  {21  -  a)  /I  (2) 

Substituting  I  from  (1)  in  (2),  we  get: 

635  =  128  a  -  a 
whence: 

a  =  5,  hence,  I  =  320* 

(4)  The  fourth  term  of  a  geometrical  progression  is  4,  and  the 
sixth  term  is  1.     What  is  the  tenth  term? 

Here: 

ar^  =  4  (1) 

and: 

ar^  =  1  (2) 


§122]  PROGRESSIONS  205 

whence,  dividing  (2)  by  (1): 

r2  =  1  /4,  or  r  =  ±  1  /2 
therefore,  from  (1) : 

a  =  4/7-3  =   +  32 
Then  the  tenth  term  is: 

+  32(  +  1/2)9=  1/16 

Exercises 

1.  Find  the  sum  of  seven  terms  of  4  +  8  +  16  +  .    .    . 

2.  Find  the  sum  of  —  4  +  8  —  16  +  .    .    .  to  six  terms. 

3.  Find  the  tenth  term  and  the  sum  of  ten  terms  of  4  —  2  + 
1  -  .    .    . 

4.  Find  r  and  s;  given  a  =  2,  I  =  31,250,  n  =  7. 

6.  Insert  two  geometrical  means  between  47  and  1269. 

6.  Insert  three  geometrical  means  between  2  and  3. 

7.  Insert  seven  geometrical  means  between  a^  and  b^. 

8.  Show   that   the    quotient    (o"  —  b'')/{a  —  &)  is  a   geometrical 
progression. 

9.  Sum  a;"  ~  1  +  x"  ~  2  2/  _|_  ^n  -  3  ^2  _|_  .  to  n  terms. 

10.  Sum  a;"  ~  1  —  X"  ~  2  2^  _j_  ^.n  -  3  ^2  _  .  to  n  terms. 

11.  Sum  a+ar~^-{-ar~^+.    .    .  to  n  terms. 

12.  If  a,  b,  c,d,  .  .  .  are  in  geometrical  progression,  then  a^  +  b^, 
b^  +  c2,  c^  -\-  d^,   .    .    .  are  also  in  geometrical  progression. 

13.  If  any  numbers  are  in  geometrical  progression,  their  differences 
are  also  in  geometrical  progression. 

14.  A  man  agreed  to  pay  for  the  shoeing  of  his  horse  as  follows: 
1  cent  for  the  first  nail,  2  cents  for  the  second  nail,  4  cents  for  the  third 
nail,  and  so  on  until  the  eight  nails  in  each  shoe  were  paid  for.  What 
did  the  last  nail  cost?     How  much  did  he  agree  to  pay  in  all? 

122.  Compound  Interest.  Just  as  the  amount  of  principle  and 
interest  of  a  sum  of  money  at  simple  interest  for  n  years  is  ex- 
pressed by  the  (n  +  l)st  term  of  an  arithmetical  progression,  so, 
in  the  same  way,  the  amount  of  any  sum  at  compound  interest  for 
n  years  is  represented  by  the  (n  +  l)st  term  of  a  geometrical  pro- 
gression. Thus,  the  amount  of  $1.00  at  compound  interest  at 
4  percent  for  twenty  years  is  given  by  the  expression: 

1(1.04)20 

The  amount  of  d  dollars  for  n  years  at  r  percent  is: 


206        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§123 

The  present  value  of  $1.00,  due  twenty  years  hence,  estimating 
compound  interest  at  4  percent,  is: 

1/(1.04)20 

The  value  of  $1.00,  paid  annually  at  the  beginning  of  each  year 
into  a  fund  accumulating  at  4  percent  compound  interest,  is,  at 
the  end  of  that  period : 

(1.04)1+  (1.04)2+    .    .    .    (1.04)20 

which  is  the  sum  of  the  terms  of  a  geometrical  progression  of 
twenty  terms. 

Problems  of  this  character  in  compound  interest  and  in  com- 
pound discount,  and  the  more  complicated  problems  that  proceed 
therefrom,  are  basal  to  the  theory  of  annuities,  life  insurance  arid 
depreciation  of  machinery  and  structures.  The  computation  of 
the  high  powers  involved  necessitates  the  postponement  of  such 
problems  until  the  subject  of  logarithms  has  been  explained. 

123.  Infinite  Geometrical  Progressions.  If  the  ratio  of  a 
geometrical  progression  be  a  proper  fraction,  the  progression  is 
said  to  be  a  decreasing  progression.     Thus: 

1,  1/2,  1/4,  1/8,  1/16^  and  1/3,  1/9,  1/27,  1/81 
are  decreasing  progressions.  If  we  increase  the  number  of  terms 
in  the  first  of  these  progressions  the  sums  will  always  be  less  than  2; 
but  the  difference  2  —  s  will  become  and  remain  less  than  any 
preassigned  number.  By  definition,  2  is,  therefore,  the  limit  of 
this  sum.^  The  sum  of  n  terms  of  this  particular  progression 
should  be  written  down  by  the  student  for  a  number  of  successive 
values  for  n,  thus: 
Number  of  terms: 

1,       2,  3,  4,  5,         .  .  .  10, 

Sum:   1,1  +  1/2,1  +  3/4,1  +  7/8,  1+  15/16,.  .  .  1  +  511/512, 
The  nth  term  differs  from  2  by  only  1/2"  -  i. 

It  is  easy  to  show  that  the  sum  of  every  decreasing  geometrical 
progression  approaches  a  fixed  limit  as  the  number  of  terms 
becomes  infinite.     For,  write  the  formula: 

a  —  ar"" 

^See  definition,  §80. 


§124]  ,  PROGRESSIONS  207 

in  the  form: 

a  ar^"  ,  . 

1  —  r        1  —  r  ^  ^ 

If  we  suppose  that  r  is  a  proper  fraction  and  that  n  increases  with- 
out Hmit,  then  r"  can  be  made  less  than  any  assigned  number,  for 
the  value  of  any  power  of  a  proper  fraction  decreases  as  the  ex- 
ponent of  the  power  increases.  As  the  other  parts  of  the  second 
fraction  in  (1)  do  not  change  in  value  as  n  changes,  the  fraction 
as  a  whole  can  be  made  smaller  than  any  number  that  can  be 
assigned.     Hence,  we  write: 

li»    s=,^  (2) 

n  =  c»  1  —  r  ^  ^ 

Exercises 

As  n  =  CO  J  find  the  limit  of  each  of  the  following: 

1.  1/2  -  1/4  +  1/8  -  1/16  +     .      .      . 
Here: 

a  =  1/2,   r  =  -  1/2 
1/2 
whence,  the  limit  s  =   -TZTTZIY/^)  ^  •^Z^- 

2.  0.3333   .    .    . 

Here:  a  =  3/10,  r  =  1/10 

whence,  the  limit:  s  =  i~_  jtvq  =  1/3. 

3.  9  -6  +4-   .    .    . 

4.  0.272727   .    .    . 

5.  0.279279279   .    .    . 

6.  1/3  -  1/6  +  1/12  -    .    .    . 

7.  4  -I-  0.8  +  0.16  +   .    .    . 

8.  Express  the  number  8  as  the  sum  of  an  infinite  geometrical 
progression  whose  second  term  is  2. 

124.  Graphical  representation  of  the  terms  and  of  the  sum  of  a 
geometrical  progression:  If  lines  proportional  to  the  terms 
of  an  arithmetical  progression  be  erected  at  equal  intervals  normal 
to  any  line,  the  ends  of  the  perpendiculars  will  he  on  a  straight 
line,  as  already  explained  in  §117.  We  shall  now  explain 
a  corresponding  construction  for  a  geometrical  progression. 
First,  note  that  all  the  essentials  of  a  geometrical  progression  may 


208        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§124 

be  studied  if  we  assume  the  first  term  to  be  unity,  for  the  number 
a  occurs  only  as  a  single  constant  multiplier  in  each  term,  and 
also  occurs  in  the  same  manner  in  the  formulas  for  I  and  s.  There- 
fore, by  taking  a-fold  these  expressions  in  a  geometrical  series 
whose  first  term  is  1,  the  results  are  obtained  for  the  more 
general  case. 

To  represent  the  geometrical  series  l+r-f-r'^-|-r^+.  .  .  + 
r--i  graphically,  lay  off  OM  =  1  on  OY,  OSi  =  1  on  OX,  SiPi  = 
r  on  the  unit  line,  and  draw  MPi.     Draw  the  arc  P1S2  and  erect 


Y 

U 

P^^^-^-^ 

— ' 

Pa   ^^^ 1 

"^^--..^^ 

■""^ 

-1 

^\^ 

^1^^^ 

N. 

r& 

M^^ 

^ 

rs! 

\ 

rs           \ 

r*               \ 

1 

'      \ 

\ 

X 

O         Si  Sa  S3  Si  Sb 

Fig.  87. — Graphical  Construction  of  the  Sum  o(  a.  G.  P.  r  >  1. 


P2*Sf2.  Draw  the  arc  P2S2  and  erect  PiS».  Continue  this  con- 
struction until  you  draw  the  arc  Pn-iSn  and  erect  PnSn.  The 
series  of  trapezoids  OMSiPi,  S1P1P2S2,  S2P2P3S3,  .  .  .  , 
Sn-iPn-iPnSn  arc  slmllar  and,  since  PiSi  =  r  XOM,  it  follows 
that  P2S2  =  rPiSi,  PsSz  =  rP2S2,  .  .  •  ,  PnSn  =  rPr.-iSn-i. 
Hence  we  have: 

OM  =  OSi    =1 

PiSi  =  S1S2  =  r 

P2S2  =  S2Ss  =  r-' 

PsSz  =  >S3^4  =  r' 


082=  1  +  r  = 

OaSs  =  1  +  r  +  r2   = 

OSi  =  1  +r  +  r2  +  r= 


sum  of  2  terms 

sum  of  3  terms 

=    sum  of  4  terms 


:.OSn  =  1  +   r-\-r^  + 


j.n-1  = 


Pn-lSn-l   =  Sn-lSn  =  T' 

sum  of  n  terms. 

Fig.  87  shows  the  series  whose  ratio  is  r  =  1.2.  Fig.  88  shows 
the  series  whose  ratio  is  0.8. 

The  line  MPi  has  the  slope  (r  —  1)  in  Fig.  87  and  the  slope 
—  (1  —  r)  in  Fig.  88.     In  both,  its  F-intercept  is  1.     Its  equation 

1  -r 


is,  in   both   cases, 


{r  —  l)x  -{-  1   or  a;  = 


In   both 


§124] 


PROGRESSIONS 


209 


figures,  when  y  =  PnSn  =  r«,  x  =  OSn.    Substituting  these  values 

1  —  r" 
for  X  and  y,  we  get  for  the  sum  of  n  terms,  S  =  ^Z. -^^S- 

87  shows  that  when  the  number  of  terms  is  allowed  to  increase 
without  limit,  the  sum  OSn  also  increases  without  limit.     Fig. 

88  shows  that  when  the  number  of  terms  is  made  to  increase 
without  limit,  the  sum  OSn  approaches  OL  as  a  limit.  Now  the 
value  of  OL  is  the  value  of  x  when  y  =  0.     Hence  the  limit 

1 


of  the  sum  of  the  progression,  or  OL  = 


1  -  r 


13,  14. 


Consult  also  §7,  problem  6,  exercise  5  and  Figs. 

In  Figs.  87  and  88  the  ordinates  OM,  Si  Pi,  S2P2,  •  .  •  repre- 
senting the  successive  terms  of  the  geometrical  progressions,  were 
not  erected  at  equal  intervals  along  OX.  If  the  ordinates  repre- 
senting the  successive  terms  of  the  progressions  be  erected  at  equal 
intervals  along  OX,  the  line  MPiP^Pz  .  .  .  passing  through 
the  ends  of  the  ordinates  will  be  a  curve  and  not  a  straight  line. 


Y 

U 

M 

Pi 

1        . 

r^ 

Pt 

Pz 

P4 

Pf> 

^ 

.  ^ 

r^N 

ri 

nJTT"     --, 

— ~-^ 

Fig.  88. — Graphical  Construction  of  the  Sum  of  a  G.  P.  r  <  1. 

To  construct  this  curve,  a  geometrical  construction  different  from 
that  given  above  is  to  be  preferred.  Near  the  lower  margin  of  a 
sheet  of  8|  X  11-inch  unruled  paper  lay  off  a  uniform  scale  of 
inches  and  draw  vertical  lines  through  the  points  of  division,  as 
shown  in  Fig.  89.  Select  one  of  these  for  the  2/-axis,  and  on  the 
unit  line  lay  off  the  given  ratio  of  the  progression  IN  =  r.  Then 
divide  the  2/-axis  proportionally  to  the  successive  powers  of  r, 
either  by  the  method  of  problem  6,  §7,  Fig.  11,  or  by  the 
method  shown  in  Fig.  89.  Through  the  points  of  division  on  the 
2/-axis  draw  lines  parallel  to  the  x-axis,  thus  dividing  the  plane 
into  a  large  number  of  rectangles.  Starting  at  the  point  M 
(0,   1)  sketch  free   hand  the  diagonals  of   successive  cornering 

14 


210        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§124 

rectangles,  rounding  the  results  into  a  smooth  curve  as  shown. 
Then  the  relation  between  ordinate  y  and  abscissa  x  for  the  values 
of  X  =  —  2,  —1,  0,  1,  2,  3,  etc.,  is  given  by  the  equation  y  =  r*. 
Fig.  89  is  drawn  f or  r  =  3  /2  so  that  the  curve  is  ?/  =  (3  /2)  *. 

The  method  used  in  Fig.  89  may  be  explained  as  follows: 
Draw  the  lines  y  =  x  and  y  =  rx.  From  the  point  (1,  r) 
on  y  =  rx  draw  a  horizontal  line  to  y  =  x,  thence  a  vertical  line 


8 

7- 
6 

.Y 

V 

*  / 

7 

/   /s 

A 

4- 

q/ 

7 

At 

7 

3- 

p/ 

'// 

/ 

2 

N 

/ 

^/ 

>r^ 

M 

^^ 

/ 

lr2 

^^^."^ 

A/ 

r 

r " 

/y 

"     ' 

P- 

7^ 

u 

Fig.  89. — Graphical  Construction  of  the  Successive  Terms  of  a  G.  P. 
In  the  diagram  r  =  3/2,  and  the  curve  is  y  =  (3/2)^. 


to  y  =  rx,  etc.,  thereby  forming  the  ''stairway"  of  line  segments 
between  y  =  x  and  y  —  rx  as  shown  in  the  figure.  Then  the 
points,  N,  P,  Q,  etc.,  have  the  ordinates  r,  r^  r^,  etc.,  as  required, 
for,  to  obtain  the  ordinate  of  P,  or  PD,  the  value  of  x  used  was 
OD  =  r,  hence  P  is  the  point  on  y  =  rx  for  x  =  r,  or  y  = 
PD  —  r^.  Likewise  Q  is  by  construction  the  point  on  y  —  rx 
for  X  =  7-2^  hence  the  y  of  the  point  Q  =  r  X  r^  =  r^  etc. 

The  figure  shows  the  process  for  finding  r~^  f^,  etc.  In 
Chapter  VIII  a  method  will  be  explained  for  locating  intermediate 
points  on  the  curve. 


§125]  PROGRESSIONS  211 

The  curve  generated  by  the  method  described  above  is  one  of 
the  most  important  curves  in  mathematics.  In  general,  it  is  seen 
that  the  points  located  on  the  curve  MN  always  satisfy  an 
equation  of  the  form 

y    =   fx 

where  r  is  a  constant.  This  is  called  an  exponential  equation 
and  the  curve  is  known  as  the  exponential  or  compoimd  interest 
curve. 

Note  that  the  ordinates  y  to  the  right  of  M  increase  rapidly  as  x 
increases  and  that  the  ordinates  to  the  left  of  M  decrease  very 
slowly  as  x  decreases;  that  is,  the  curve  rapidly  leaves  the  positive 
a: -axis,  but  slowly  approaches  the  negative  x-axis  as  an  asymp- 
tote.    These  results  are  exactly  reversed  in  case  r  <  1. 

125.  *  Harmonical  Progressions.  A  series  of  terms  such  that 
their  reciprocals  form  an  arithmetical  progression  are  said  to  form 
an  harmonical  progression.     The  following  are  examples: 

(1)  1/2,  1/3,  1/4,  1/5. 

(2)  1,  1/5,  1/9,  1/13. 

(3)  ll{x-y),  llx,ll{x-\-y). 

(4)  1/3,1,  -1,  -1/3. 

(5)  4,  6,  12. 

(6)  1/a,  l/(aH-d),  l/(a  +  2d),   .    .    . 

Although  harmonical  progressions  are  of  such  a  simple  character, 
no  simple  expression  has  been  found  for  the  sum  of  n  terms.  Our 
knowledge  of  arithmetical  progressions  enables  us  to  find  the 
value  of  any  required  term  and  to  insert  any  required  number 
of  harmonical  means  between  two  given  extremes,  as  in  the 
examples  below. 

( 1)  Write  six  terms  of  the  harmonical  progression  6,  3,  2. 

We  must  write  six  terms  of  the  arithmetical  progression, 
1/6,  1/3,  1/2.  The  common  difference  of  the  latter  is  1/6,  so 
that  the  arithmetical  progression  is  1  /6,  1  /3,  1  /2,  2  /3,  5/6,  1 ,  and 
the  harmonical  progression  is  6,  3,  2,  1.5,  1.2,  1. 

(2)  Insert  two  harmonical  means  between  4  and  2. 

We  must  insert  two  arithmetical  means  between  1/4  and  1/2; 
these  are  1/3  and  5/12,  whence  the  required  harmonical  means 
are  3  and  2.4. 


212        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§126 

126.*  Hannonical  Mean.  The  harmonical  mean  is  found  as 
follows:  Let  the  two  numbers  be  a  and  h  and  let  H  stand  for  the 
required  mean.     Then  we  have: 

IfH  -  l/a  =  1/6  -  1/H 


That  is: 


whence: 


2IH  =  l/a+  1/6  =  (a  +  b) /ab 


H  =  2ab/(a  +  b)  (1) 

Thus  the  harmonical  mean  of  4  and  12  is  96/(4  +  12)  =  6. 
By  the  harmonical  mean  of  several  numbers  is  meant  the  reciprocal 
of  the  arithmetical  mean  of  their  reciprocals.  Thus  the  har- 
monical meaui  of  12,  8  and  48  is  13tt. 


Fig.  90. — The  Relation  Between  the  Arithmetical,  Geometrical  and  Har- 
monic Means. 

127.*  Relation  between  A,  G,  and  H.     As  previously  found: 

A  =  (a  +  6)  /2,  G  =  ^Jab^H  =  2a6/(a  +  b) 
whence: 

AH  =^  ab 
but: 

ab  =  G^ 
hence: 

AH  =  G^ 
or: 


G  =   VAH 


(1) 


§127]  PROGRESSIONS  213 

That  is  to  say,  the  geometrical  mean  of  any  two  positive  numbers 
is  the  same  as  the  geometrical  mean  of  their  arithmetical  and 
harmonical  means. 

The  arithmetical,  geometrical  and  harmonical  means  may  be 
constructed  graphically  as  in  Fig.  90.  Draw  the  circle  of  diameter 
{a  -\-  b)  =  OM  +  MK.  Then  the  radius  is  the  arithmetical 
mean  A.  Erect  a  perpendicular  at  M.  Then  MG  is  the  geomet- 
rical mean.  MakeOG'  =  MG  and  draw  CG'.  Draw  G'H  perpen- 
dicular to  CG\     Then  OH  is  the  harmonical  mean,  since 


OG'  =  sloe  X  OH 

Now  A  >G  >  H'Jor  from  the  figure,  MG  <  CA.  Therefore, 
the  angle  G'CO  is  less  than  45°  and  also  its  equal  HG'O  is  less 
than  45°.  Therefore,  HO  <  OG'  which  estabhshes  the  in- 
equahty. 

Exercises 

1.  Continue  the  harmonical  progression  12,  6,  4. 

2.  Find  the  difference:     (1.8  +  1.2  +  0.8  +    .    .    .   to  8  terms) 

-  (1.8  +  1.2  +  0.6  +    ...   to  8  terms). 

3.  If  the  arithmetical  mean  between  two  numbers  be  1,  show  that 
the  harmonical  mean  is  the  square  of  the  geometrical  mean. 


CHAPTER  VIII 

THE  LOGARITHMIC  AND  THE  EXPONENTIAL 
FUNCTIONS 

128.  Historical  Development.  The  almost  miraculous  power 
of  modern  calculation  is  due,  in  large  part,  to  the  invention  of 
logarithms  in  the  first  quarter  of  the  seventeenth  century  by  a 
Scotchman,  John  Napier,  Baron  of  Merchiston.  This  invention 
was  founded  on  the  simplest  and  most  obvious  of  principles,  that 
had  been  quite  overlooked  by  mathematicians  for  many  genera- 
tions. Napier's  invention  may  be  explained  as  follows :  ^  Let  there 
be  an  arithmetical  and  a  geometrical  progression  which  are  to  be 
associated  together,  as,  for  example,  the  following: 

0,  1,     2,     3,      4,       5,       6,       7,        8,        9,         10 

1,  2,  4,  8,  16,  32,  64,  128,  256,  512,  1024 
Now  the  product  of  any  two  numbers  of  the  second  line  may  be 
found  by  adding  the  two  numbers  of  the  first  progression  above 
them,  finding  this  sum  in  the  first  line,  and  finally  taking  the  num- 
ber lying  under  it;  this  latter  number  is  the  product  sought.  Thus, 
suppose  the  product  of  8  by  32  is  desired.  Over  these  numbers 
of  the  second  line  stand  the  numbers  3  and  5,  whose  sum  is  8. 
Under  8  is  found  256,  the  product  desired.  Now  since  but  a 
limited  variety  of  numbers  is  offered  in  this  table,  it  would  be 
useless  in  the  actual  practice  of  multiplication,  for  the  reason 
that  the  particular  numbers  whose  product  is  desired  would 
probably  not  be  found  in  the  second  line.  The  overcoming 
of  this  obvious  obstacle  constitutes  the  novelty  of  Napier's  inven- 
tion. Instead  of  attempting  to  accomplish  his  purpose  by  ex- 
tending the  progressions  by  continuation  at  their  ends,  Napier 
proposed  to  insert  any  number  of  intermediate  terms  in  each 
progression.     Thus,  instead  of  the  portion 

0,  1,     2,     3,      4 

1,  2,    4,     8,     16 
of  the  two  series  we  may  write: 

1  Merely  the  fundamental  principles  of  the  invention,  not  historical  details,  are 
given  in  what  follows. 

214 


§1281  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   215 

0.  1/2,       1,  Ih      2,       2i         3,  3i         4 

1,  V'2,       2,       \/8,      4,     V32,       8,   \/l28,       16 

by  inserting  arithmetical  means  between  the  consecutive  terms 
of  the  arithmetical  series  and  by  inserting  geometrical  means 
between  the  terms  of  the  geometrical  series.  Let  these  be 
computed  to  any  desired  degree  of  approximation,  say  to  two 
decimal  places.     Then  we  have  the  series 


A.P. 

G.P. 

0.0 

1.00 

0.5 

1.41 

1.0 

2.00 

1.5 

2.83 

2.0 

4.00 

2.5 

5.66 

3.0 

8.00 

imetical  and  ^ 

;eometric{ 

,ive  series  we 

have: 

A.  P. 

G.P. 

0.00 

1.00 

0.25 

1.19 

0.50 

1.41 

0.75 

1.69 

1.00 

2.00 

1.25 

2.38 

1.50 

2.83 

1.75 

3.36 

2.00 

4.00 

2.25 

4.76 

By  continuing  this  process  each  consecutive  three  figure  number 
may  finally  be  made  to  appear  in  the  second  column,  so  that,  to 
this  degree  of  accuracy,  the  product  of  any  two  such  numbers 
may  be  found  by  the  process  previously  explained.  The  decimal 
points  of  the  factors  may  be  ignored  in  this  work,  as  for  example, 
the  product  of  2.38  X  14.1  is  the  same  as  that  of  238  X  14.1 
except  in  the  position  of  the  decimal  point.  The  correct  position 
of  the  decimal  point  can  be  determined  by  inspection  after  the 


216        ELEMENTARY  MATHEMATICAL  ANALYSIS      i§129 

significant  figures  of  the  product  have  been  obtained.  Using 
the  above  table  we  find  2.38  X  14.1  =  33.6. 

The  above  table,  when  properly  extended,  is  a  table  of  loga- 
rithms. As  geometrical  and  arithmetical  progressions  different 
from  those  given  above  might  have  been  used,  the  number  of 
possible  systems  of  logarithms  is  indefinitely  great.  The  first 
column  of  figures  contains  the  logarithms  of  the  numbers  that 
stand  opposite  them  in  the  second  column.  Napier,  by  this 
process,  said  he  divided  the  ratio  of  1.00  to  2.00  into  ''100  equal 
ratios,"  by  which  he  referred  to  the  insertion  of  100  geometrical 
means  between  1.00  and  2.00.  The  "number  of  the  ratio" 
he  called  the  logarithm  of  the  number,  for  example,  0.75  opposite 
1.69,  is  the  logarithm  of  1.69.  The  word  logarithm  is  from  two 
Greek  words  meaning  ^'  The  number  of  the  ratios."  In  order  to 
produce  a  table  of  logarithms  it  was  merely  necessary  to  compute 
numerous  geometrical  means;  that  is,  no  operations  except  multi- 
plication and  the  extraction  of  square  roots  were  required.  But 
the  numerical  work  was  carried  out  by  Napier  to  so  many  decimal 
places  that  the  computation  was  exceedingly  difficult. 

The  news  of  the  remarkable  invention  of  logarithms  induced 
Henry  Briggs,  professor  at  Gresham  College,  London,  to  visit 
Napier  in  1615.  It  was  on  this  visit  that  Briggs  suggested  the  ad- 
vantages of  a  system  of  logarithms  in  which  the  logarithm  of 
1  should  be  0  and  the  logarithm  of  10  should  be  1,  for  then  it  would 
only  be  necessary  to  insert  a  sufficient  number  of  geometrical 
means  between  1  and  10  to  get  the  logarithm  of  any  desired 
number.  With  the  encouragement  of  Napier,  Briggs  undertook 
the  computation,  and  in  1617,  pubHshed  the  logarithms  of  the 
first  1000  numbers  and,  in  1624,  the  logarithms  of  numbers  from 
1  to  20,000,  and  from  90,000  to  100,000  to  fourteen  decimal 
places.  The  gap  between  20,000  and  90,000  was  filled  by  a  Hol- 
lander, Adrian  Vlacq,  whose  table,  published  in  1628,  is  the  source 
from  which  nearly  all  the  tables  since  published  have  been 
derived. 

129.  Graphical  Computation  of  Logarithms.  In  Fig.  89  the 
terms  of  a  geometrical  progression  of  first  term  1  and  ratio  lA^  =  r 
are  represented  as  ordinates  arranged  at  equal  intervals  along  OX. 
Fig.  89  is  drawn  to  scale  for  the  value  of  r  =  1.5.     Fig.  91  is 


§129]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   217 


a  similar  figure  drawn  for  r  =  2,  in  which  a  process  is  used  for 
locating  intermediate  points  of  the  curve,  so  that  the  locus  may 
be  sketched  with  greater  accuracy.  The  lines  y  =  x  and  y  =  rx 
(in  this  case  y  =  2x)  are  drawn  as  before,  and  the  ''stairway" 
constructed  as  before  (see  §124).  Vertical  lines  drawn 
through  X  =  —  2,  —  1,  0,  1,  2,3,  .    .    .  and  horizontal  lines  drawn 


9 

Y 

U 

/, 

e 

/ 

i 

^ 

f 

f 

/ 

7- 
6- 

/ 

/ 

/ 

/ 

/ 

/ 

< 

y 

<B 

5. 
4 

/ 

/ 

/ 

/ 

C 

/ 

/ 

/ 

3. 

I' 

/ 

/ 

/ 

/ 

?, 

N 

^ 

/ 

B 

A 

/ 

/ 

/ 

/ 

M 

/ 

// 

A 

= 

— 

i 

— 

4 

^ 

1 

i 

= 

i 

— 

~ 

= 

^ 

= 

::3: 

-2-1012345  X 

Fig.  91. — Graphical  Construction  of  the  Curve  'y  =  2*. 

through  the  horizontal  tread  of  each  step  of  the  stairway  divides 
the  plane  into  a  large  number  of  rectangles.  Starting  at  M 
and  sketching  the  diagonals  of  successive  cornering  rectangles 
the  smooth  curve  MiVP  is  drawn.  Intermediate  points  of 
the  curve  are  located  by  doubling  the  number  of  vertical  lines  by 
bisecting  the  distances  between  each  original  pair,  and  then 
by  increasing  the  number  of  horizontal  lines  in  the  following  man- 
ner: Draw  the  Hne  ?/  =  Vrx  (in  the  case  of  the  Fig.,  y  —  V2  x). 


218        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§129 

At  the  points  where  this  line  cuts  the  vertical  risers  of  each  step 
of  the  '' stairway '^  (some  of  these  points  are  marked  A,  B,  C 
in  the  diagram)  draw  a  new  set  of  horizontal  lines.  Each  of  the 
original  rectangles  is  thus  divided  into  four  smaller  rectangles. 
Starting  at  M  and  sketching  a  smooth  curve  along  the  diagonals 
of  successive  cornering  rectangles,  the  desired  graph  is  obtained. 
By  the  use  of  the  straight  line  y  =  Mr  x  another  set  of  intermedi- 
ate points  may  be  located,  and  so  on,  and  the  resulting  curve 
thus  drawn  to  any  degree  of  accuracy  required.  In  explaining 
this  process,  the  student  will  show  that  the  method  of  construc- 
tion just  used  consists  in  the  doubling  of  the  number  of  horizontal 
lines  of  the  figure  by  the  successive  insertion  of  geometrical  means 
between  the  terms  of  a  geometrical  progression,  while  at  the  same 
time  the  number  of  vertical  lines  is  successively  doubled  by 
insertion  of  arithmetical  means  between  the  terms  of  an  arith- 
metical series.  Thus  the  graphical  work  of  construction  of  the 
curve  corresponds  to  the  successive  insertion  of  geometrical  and 
arithmetical  means  in  the  two  series  discussed  in  the  preceding 
section. 

As  explained  above,  the  ordinate  y  of  any  point  of  the  curve 
MNP  of  Fig.  91  is  a  term  of  a  geometrical  progression,  and  the 
abscissa  x  of  the  same  point  is  the  corresponding  term  of  an 
arithmetical  progression.  Since,  when  y  is  given,  the  value  of  x 
is  determined,  we  say,  by  definition,  that  x  is  a  junction  of  y 
(§4).  This  particular  functional  relation  is  so  important 
that  it  is  given  a  special  name:  x  is  called  the  logarithm  of  y, 
and  the  statement  is  abbreviated  by  writing 

X  =  \ogy, 

but  to  distinguish  from  the  case  in  which  some  other  geometrical 
progression  might  have  been  used,  the  ratio  of  the  progression 
may  be  written  as  a  subscript,  thus: 

X  =  logr  y 

which  is  read:  "a:  is  the  logarithm  of  y  to  the  base  r." 

If  we  assume  that  the  process  of  locating  the  successive  sets  of 
intermediate  points  by  the  construction  of  successive  geometrical 
means  will  lead,  if  continued  indefinitely,  to  the  generation  of 


§129]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   219 

the  curve  MNP  without  breaks  or  gaps,  then  we  may  say  that  in 
the  equation: 

x  =  logry  (1) 

the  logarithm  is  a  function  of  y  defined  for  all  positive  values  of  y 
and  for  all  values  of  x. 

As  a  matter  of  fact,  both  the  arithmetical  and  the  geometrical 
method  given  above  defines  the  function  or  the  curve  only  for 
rational  values  of  x ;  that  is,  the  only  values  of  x  that  come  into 
view  in  the  process  explained  above  are  whole  numbers  and 
intermediate  rational  fractions  like  2|,  2^,  2f,  2^^,  2jf ,   .    .    . 

It  is  seen  at  once  from  the  method  of  construction  used  in  Fig. 
91  that  the  values  of  ?/  at  x  =  1,  2,  3,  4,  .  .  .,  are  respectively 
y  =  r,  r^,  r^,  r'*,  .  .  . ,  and  the  values  oi  y  atx  =  1/2,  3/2,  5/2,  .  .  . , 
are  y  =  r^'^\  r^^,  r^^,  .  .  . ,  respectively,  and  similarly  for  other  inter- 
mediate values  of  x.  In  other  words,  the  equation  connecting 
the  two  variables  x  and  y  may  be  written 

y  =  r-  (2) 

Thus,  when  the  values  of  a  variable  x  run  over  an  arithmetical 
progression  {of  first  term  0)  while  the  corresponding  values  of  a 
variable  y  run  over  a  geometrical  progression  {of  first  term  1),  the 
relation  between  the  variables  may  be  written  in  either  of  the  forms 
(1)  or  (2)  above.  Equation  (2)  is  called  an  exponential  equation 
and  y  is  said  to  be  an  exponential  function  of  x,  while  in  (1)  x 
is  said  to  be  a  logarithmic  f miction  of  y.  The  student  has  fre- 
quently been  called  upon  in  mathematics  to  express  relations 
between  variables  in  two  different  or  ''inverse"  forms,  analogous 
to  the  two  forms  y  =  r''  and  x  =  logr  y.  For  example,  he  has 
written  either 

2/  =  x2 
or: 

X  =   ±  vV 
and  either 


X    =   ?/2/n 

The  graph  of  a  function  is  of  course  the  same  whether  the  equation 
be  solved  for  x  or  solved  for  y. 


220        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§130 

130.  The  student  is  required  to  construct  the  curves  described 
in  the  following  exercises  by  the  method  of  §129.  The 
inch,  or  2  cm.,  may  be  adopted  as  the  unit  of  measure;  the  curves 
should  be  drawn  on  plain  paper  within  the  interval  from  x  = 
-  2  to  a;  =  +  2. 

If  tangents  be  drawn  to  the  curves  at  a;  =  —  2,  —  1,  0,  1,  2, 
it  will  be  noted,  as  nearly  as  can  be  determined  by  experiment, 
that  the  several  tangents  to  any  one  curve  cut  the  X-axis  at  the 
same  constant  distance  to  the  left  of  the  ordinate  of  the  point 
of  tangency.  This  distance  is  greater  than  unity  if  r  =  2  and  less 
than  unity  if  r  =  3.  The  value  of  r  for  which  the  distance  is  exactly 
unity  is  later  shown  to  be  a  certain  irrational  or  incommensurable 
number,  approximately  2.7183  .  .  .  ,  represented  in  mathematics 
by  the  letter  e,  and  called  the  Naperian  base.  This  number,  and 
the  number  tt,  are  two  of  the  most  important  and  fundamental 
constants  of  mathematics.^ 

1  It  is  not  easy  to  locate  accurately  the  tangent  to  a  curve  at  a  given  point 
of  the  curve.  To  test  whether  or  not  a  tangent  is  correctly  drawn  at  a  point 
P,  a  number  of  chords  parallel  to  the  tangent  may  be  drawn.  If  the  two  end 
points  AB  of  the  chord  tend  to  approach  the  point  of  tangency  P  as  the  chord 
is  taken  nearer  and  nearer  to  P  (but  always  parallel  to  AB)  then  the  tangent 
was  correctly  drawn.  If  the  two  points  A  and  B  do  not  tend  to  coalesce  at  the 
point  P  when  the  chord  is  moved  in  the  manner  described,  then  the  tangent 
was  incorrectly  drawn. 

A  number  of  instruments  have  been  designed  to  assist  in  drawing  tangents  to 
curves.      One  of  these,  called  a  "Radiator,"  will  be  found  listed  in  most  catalogs 


Fig.  92. — Mirrored  Ruler  for  Drawing  the  Normal  (and  hence  the  Tan- 
gent) to  any  Curve. 

of  drawing  instruments.  Another  instrument  consists  of  a  straight  edge  provided 
with  a  vertical  mirror  as  shown  in  Fig.  92.  When  the  straight  edge  is  placed 
across  a  curve  the  reflection  of  the  curve  in  the  mirror  and  the  curve  itself  can 
both  be  seen  and  usually  the  curve  and  image  meet  to  form  a  cusp  or  angle. 
The  straight  edge  may  be  turned,  however,  until  the  image  forms  a  smooth 
continuation  of  the  given  curve.  In  this  position  the  straight-edge  is  perpendicu- 
lar to  the  tangent  and  the  tangent  can  then  be  accurately  drawn.  See  Gram- 
berg,  Technische  Messungen,  1911. 


§131]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   221 

Exercises 

Draw  the  following  curves  on  plain  paper  using  1  inch  as  the  unit 
of  measure;  make  the  tests  referred  to  in  the  second  paragraph  of 
§130. 

1.  Construct  a  curve  similar  to  Fig.  91,  representing  the  equation 
X  =  log2  y,  from  X  =  — 2toa;  =  +2,  and  draw  tangents  at  a;  =  —  1, 
X  =  0,  a;  =  1,  X  =  2. 

2.  Construct  the  curve  whose  equation  is  x  =  logs  y  from  x  =  —  2 
to  X  =  +2,  and  draw  tangents  at  x  =  —  1,  x  =  0,  x  =  1,  x  =  2. 

3.  Construct  the  curve  whose  equation  is  a:  =  log2.7  y,  and  show  by 
trial  or  experiment  that  the  tangent  to  the  curve  at  a;  =  2  cuts  the  x-axis 

,at  nearly  a;  =  1,  that  the  tangent  at  x  =  1  cuts  the  a;-axis  at  nearly 
X  =  0,  that  the  tangent  at  a;  =  0  cuts  the  x-axis  at  nearly  a:  =  —  1, 
etc. 

4.  Draw  the  curve  x  =  logo.  6  y  and  show  that  it  is  the  same  as  the 
reflection  of  a:  =  loga  y  in  the  mirror  x  =  0. 

Note:  The  student  must  remember  that  the  experimental  testing 
of  the  properties  of  the  tangents  to  the  curves  called  for  above  does  not 
constitute  mathematical  proof  of  the  usual  deductive  sort  familiar  to 
him.  The  experimental  tests  have  value,  however,  in  preparing  the 
student  for  the  rigorous  investigation  of  these  same  properties  when 
taken  up  in  the  calculus. 

131.  The  Exponential  Function.  The  expression  a^,  where  a 
is  any  positive  number  except  1,  has  a  definite  meaning  and 
value  for  all  positive  or  negative  rational  values  of  x,  for  the 
meaning  of  numbers  affected  by  positive  or  negative  fractional 
exporents  has  been  fully  explained  in  elementary  algebra.  The 
process  outlined  above  likewise  defines  log,  x  for  all  rational 
values  of  x,  but  the  process  would  not  lead  to  irrational  values 
of  X,  such  as  V2,  ^5,  etc.  As  a  matter  of  fact  the  expression  a* 
has  as  yet  no  meaning  assigned  to  it  for  irrational  values  of  x; 
thus  10  has  no  meaning  by  the  definitions  of  exponents  pre- 
viously given,  for  \/2,  is  not  a  whole  number,  hence  lO'^^  does 
not  mean  that  10  is  repeated  as  a  factor  a  certain  number  of 

times;  also  •\/2  is  not  a  fraction,  so  that  10  cannot  mean  a 
power  of  a  root  of  10.  But  if  any  one  of  the  numbers  of  the 
following  sequence 

1         1.4         1.41         1.414         1.4142         1.41421       .    .    . 


222        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§131 

be  used  as  the  exponent  of  10,  the  resulting  power  can  be  com- 
puted to  any  desired  number  of  decimal  places.  For  example, 
101-41  is  the  141th  power  of  the  lOOthroot  of  10;  to  find  the  100th 
root  we  may  take  the  square  root  of  10,  find  the  square  root  of 
this  result,  then  find  its  5th  root,  finally  finding  the  5th  root 
of  this  last  result. 
If  the  various  powers  be  thus  computed  to  seven  places  we  find: 


101-4 

=  25.11887  . 

101.41 

=  25.70396  . 

101.414 

=  25.94179  . 

101.4142 

=  25.95374. 

101  .41421 

=  25.95434  . 

101.414213 

=  25.95452. 

101.414213  5 

=  25.95455  . 

Now  the  sequence  of  exponents  used  in  the  first  column  are 
found  by  extracting  the  square  root  of  2  to  successive  decimal 
places.     If  the  sequence  in  the  second  column  approaches  a  limit, 

this  limit  is  taken  by  definition  as  the  value  of  10  .  It  is  shown 
in  higher  mathematics  that  such  a  limit  in  this  and  similar  cases 
always  exists  and  consequently  that  a  number  with  an  irrational 
exponent  has  a  meaning.  In  this  book  we  shall  assume,  without 
a  formal  proof,  that  a^  has  a  meaning  for  irrational  values  of  x. 
To  summarize:  In  order  logically  to  complete  the  definition  of 
a*  for  irrational  values  of  x,  and  to  set  forth  other  important 
properties,  we  would  be  required  to  proceed  as  follows: 

(1)  It  must  be  shown  that  if  x  be  an  always  rational  variable 
approaching  an  irrational  number  n  as  a  limit,  that  the  limit  of 
a^  exists.  The  notation  a^,  where  x  is  rational,  is  understood  to 
mean  the  positive  value  of  a'',  so  that  the  limit  of  a^,  when  it  is 
shown  to  exist,  will  necessarily  be  a  positive  number. 

(2)  The  above  described  limit  of  a^  must  be  taken  as  the  defini- 
tion of  a**,  where  n  is  the  irrational  number  approached  by  x  as 
a  limit. 

(3)  It  must  be  shown  that  a^  is  a  continuous  function  of  x. 

(4)  It  must  be  shown  that  the  fundamental  laws  of  exponents 
apply  to  numbers  affected  with  irrational  exponents. 

When  it  is  shown,  or  when  it  is  assumed,  that  a  value  of  x 


§132]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   223 

always  exists  which  will  satisfy  the  equation  a^  —  y,  where  a 
and  y  are  any  given  positive  numbers,  then  the  expression  a* 
is  called  the  exponential  function  of  x  with  base  a;  otherwise  a""  is 
defined  only  for  rational  values  of  x. 

132.  Definitions.     In  the  exponential  equation  a""  =  y: 
The  number  a  is  called  the  base. 

The  number  y  is  called  the  eicponential  function  of  x  to  the  base 
a,  and  is  sometimes  written  y  =  expa  x. 

The  number  x  is  called  the  logarithm  of  y  to  the  base  a,  and 
is  written  x  =  log  ay.     Thus  in  the  equation  a''  =  y,  x  may  be 
called  either  the  exponent  of  a  or  the  logarithm  of  y. 
The  two  equations: 

2/  =  a* 
X  =  logo  y 

express  exactly  the  same  relations  between  x  and  y;  one  equation 
is  solved  for  x,  the  other  is  solved  for  y.     The  graphs  are  identical; 
just  as  the  graphs  oi  y  =  x^  and  x  =  +  \/y  are  identical. 
See  also  Anti-logarithm,  §142. 

133.  Common  Logarithms.  In  the  equation  10"=  =  y,  x  is 
called  the  common  logarithm  of  y.  It  is  also  called  the  Brigg's 
logarithm  of  y.  Thus,  the  common  logarithm  of  any  number  is 
the  exponent  of  the  power  to  which  10  must  be  raised  to  produce 
the  given  number.  Thus  2  is  the  common  logarithm  of  100, 
since  10^  =  100;  likewise  1.3010  will  be  found  to  be  the  common 
logarithm  of  20  correct  to  4  decimal  places,  since  lO^-^o^o 
=  20.0000  to  4  decimal  places. 

134.  Systems  of  Logarithms.  If  in  the  exponential  equation 
y  =  a^,  where  a  is  any  positive  number  except  1,  different  values 
be  assigned  to  y  and  the  corresponding  values  of  x  be  computed 
and  tabulated,  the  results  constitute  a  system  of  logarithms. 
The  number  of  different  possible  systems  is  unlimited,  as  already 
noted  in  §128.  As  a  matter  of  fact,  however,  only  two 
systems  have  been  computed  and  tabulated;  the  natural  or 
Naperian  or  hyperbolic  system,  whose  base  is  an  incommensurable 
number,  approximately  2.7182818,  and  the  common  or  Briggs* 
system,  whose  base  is  10.  The  letter  e  is  set  aside  in  mathematics 
to  stand  for  the  base  of  the  natural  system. 


224        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§135 

Natural  logarithms  of  all  numbers  from  1  to  20,000  have 
been  computed  to  17  decimal  places.  The  common  logarithms 
arc  usually  printed  in  tables  of  4,  5,  6,  7  or  8  decimal  places. 

It  will  be  found  later  that  the  graphs  of  all  logarithmic  functions 
of  the  form  x  —  logo  2/  can  be  made  by  stretching  or  by  contract- 
ing in  the  same  fixed  ratio  the  ordinates  of  any  one  of  the  logarith- 
mic curves.  For  that  reason  numerical  tables  in  more  than 
one  system  of  logarithms  are  unnecessary. 

In  the  following  pages  the  common  logarithm  of  any  number  n 
will  be  written  log  n,  and  not  Icgio  n;  that  is,  the  base  is  supposed 
to  be  10  unless  otherwise  designated;  In  x  for  loge  x  and  Ig  x  for 
logiox  are  also  used. 

Exercises 

Write  the  following  in  logarithmic  notation. 

1.  103  =  1000. 

2.  10-3  ^  0.001. 

3.  100  =  1. 

4.  112  =  121. 

5.  160-25  =  2. 

6.  e^  =  y. 

7.  100-25  =  1.7783. 

8.  100.3010  =:  2. 

9.  a^  =  a. 

10.  10l°^„^        =    y. 

Express  the  following  in  exponential  notation : 

11.  logic  4  =        0.6021. 

12.  log  10000  =4. 

13.  log  0.0001  =    -  4. 

14.  Iog2l024  =10. 

15.  logo  a  =         1 . 

16.  logVlOO  =         2/3. 

17.  log27(l/3)  =  -1/3. 

18.  logioolO  =1/2. 

19.  log  1  =         0. 

20.  logol  =         0. 

135.  Graphical  Table.  In  Fig.  93  is  shown  the  graph  of  the 
function  defined  by  the  two  progressions  whose  use  was  suggested 
by  Briggs  to  Napier,  and  which  are  referred  to  in  the  last  para- 


§135]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   225 

graph  of  §128.  By  inserting  means  three  times  between  0 
and  1  in  the  arithmetical  progression  and  between  1  and  10  in  the 
geometrical  progression,  we  get 


A.  P.  or 

G.  P.  or 

Exponential 

Logarithms 

Numbers 

Form  of  G. 

0.000 

1.000 

JQO.OOO 

0.125 

1.334 

100.12.5 

0.250 

1.778 

100-250 

0.375 

2.371 

100.375 

0.500 

3.162 

100.500 

0.625 

4.217 

100.625 

0.750 

5.623 

100.750 

0.875 

7.499 

100.875 

1.000 

30.000 

101-000 

10 

N 

J 

1 

/ 

/ 

/ 

/ 

L 

=  1 

^£ 

'lO 

N 

/ 

or 

y 

r^ 

A 

r  = 

10 

L 

/ 

/ 

/ 

/ 

/ 

o 

/ 

^ 

y 

^ 

y 

^ 

.— - 

-^ 

l" 

0 

0 

1 

0 

2 

0 

3 

0 

4 

0 

5 

0 

6 

0 

7 

0 

8 

0 

9  Ll\ 

0 

Fig.  93.— The  Curve  L  =  logioiV. 


If  we  let  L  stiand  for  the  logarithm  of  the  number  iV,  the 
functional  relation  is  obviously  L  =  logioA^  or  N  =  10^.  The 
curve  (Fig.  93)  may  now  be  used  as  a  graphical  table  of  logarithms 
from  which  the  results  can  be  read  to  about  3  decimal  places. 

15 


226        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§136 

The  logarithms  of  numbers  between  1  and  10  may  be  read  directly 
from  the  graph.  Thus,  logio  7.24  =  0.860.  If  the  logarithm  is 
between  0  and  1,  the  number  is  read  directly  from  the  graph. 
Thus  if  the  logarithm  is  0.273,  the  number  is  1.87. 

If  we  multiply  the  readings  of  the  iV-scale  by  10",  we  must  add 
n  to  the  readings  on  the  L-scale,  for  10"A^  =  10^  +  ^ 

If  we  divide  the  readings  on  the  A^-scale  by  10",  we  must 
subtract  n  from  the  readings  on  the  L-scale,  for  A^  /lO"  =  10^  ~  ". 

This  fact  enables  us  to  read  the  logarithms  of  all  numbers  from 
the  graph,  and  conversely  to  find  the  number  corresponding  to 
any  logarithm.  Thus  we  have,  log  72.4  =  1.860,  log  724  =  2.860, 
log  0.724  =  0.860  -  1,  log  0.0724  =  0.860  -  2. 

If  the  logarithm  is  1.273,  the  number  is  18.7. 

If  the  logarithm  is  2.273.  the  number  is  187. 

If  the  logarithm  is  0.273  -  1,  the  number  is  0.187. 

If  the  logarithm  is  0.273  -  2,  the  number  is  0.0187. 

We  observe  that  the  computation  of  a  three  place  table  of 
logarithms  would  not  involve  a  large  amount  of  work:  such  a  table 
has  actually  been  computed  in  drawing  the  curve  of  Fig.  93. 
The  original  tables  of  Briggs  and  Vlacq  involved  an  enormous 
expenditure  of  labor  and  extraordinary  skill,  or  even  genius  in 
computation,  because  the  results  were  given  to  fourteen  places 
of  decimals. 

136.  Properties  of  Logarithms.  The  following  properties,  of 
logarithms  follow  at  once  from  the  general  properties  or  laws  of 
exponents. 

(1)  The  logarithm  of  1  is  0  in  all  systems.  For  a°  =  1,  that 
is,  logo  1  =  0.  In  Fig.  91,  note  that  the  curve  passes  through 
(0,  1). 

(2)  The  logarithm  of  the  base  itself  in  any  system  is  1.  For 
a^  =  1,  that  is,  log*  a  =  1.  In  Fig.  91,  by  construction  N  is  always 
the  point  (1,  r),  where  r  is  the  ratio  of  the  first  or  fundamental 
progression;  in  the  present  notation,  this  is  the  point  (1,  a). 

(3)  Negative  numbers  have  no  logarithms.  This  follows  at 
once  from  §131,  (1).  In  Figs.  89,  91,  and  93,  note  that  the 
curves  do  not  extend  below  the  X-axis. 

Note  :  While  negative  numbers  have  no  logarithms,  this  does  not 
prevent  the  computation  of  expressions  containing  negative  factors 


§137]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   227 

and  divisors.     Thus  to  compute  (287)  X  (—  374),  find  by  logarithms 
(287)  X  (374)  and  give  proper  sign  to  the  result. 

137.  Logarithm  of  a  Product.    Let  n  and  r  be  any  two  positive 

numbers  and  let: 

logo  n  —  X  and  logo  r  =  y  (1) 

Then,  by  definition  of  a  logarithm: 

n  =  a""  and  r  =  av  (2) 

Multiplying: 

nr  =  a^'ay  =  a*+*' 
Therefore,  by  definition  of  a  logarithm  §132: 

logo  nr  =  X  -\-  y 
or,  by  (1) 

loga  nr  =  logo  n  +  logo  r  (3) 

Hence,  the  logarithm  of  the  'product  of  two  numbers  is  equal  to 
the  sum  of  the  logarithms  of  those  numbers. 
In  the  same  way,  if  logo  s  =z ,  then: 
nrs  =  a=*+2/+« 
that  is, 

logo  nrs  =  logo  n  +  logo  r  +  logo  s 

Exercises 

Find  by  the  formulas  and  check  the  results  by  the  curve  of  Fig.  93. 

1.  Given  log  2  =  0.3010,  and  log  3  =  0.4771;  find  log  6;  find  log  18. 

2.  Given  log  5  =  0.6990  and  log  7  =  0.8451;  find  log  35. 

3.  Given  log  9  =  0.9542,  find  log  81. 

4.  Given  log  386  =  2.5866  and  log  857  =  2.9330;  find  the  logarithm 
of  the  product. 

6.  Given  log  11a;  =  1.888  and  log  11  =  1.0414;  find  log  x. 

138.  Logarithm   of  a    Quotient.    Let   n  and  r  be    any   two 
positive  numbers,  and  let: 

logo^i  =  X  and  logor  =  y  (1) 

From  (1)  by  the  definition  of  a  logarithm, 

n  =  a''  r  =  a" 

Dividing, 

n  Ir  =  a'  -7-  a"  ==  a""  -  y 


228        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§139 

Therefore  by  definition  of  a  logarithm, 

log  a(w/r)  =  x  -  2/ 
or  by  (1) 

logo(n/r)  =  loga  n  -  loga  r  (2) 

therefore,  the  logarithm  of  the  quotient  of  two  numbers  equals  the 
logarithm  of  the  dividend  less  the  logarithm  of  the  divisor. 

Exercises 

Check  the  results  by  reading  them  off  the  curve  of  Fig.  93. 

1.  Given  log  5  =  0.6990  and  log  2  =  0.3010;  find  log  (5/2);  find 
log  0.4. 

2.  Given  log  63  =  1.7993,  and  log  9  =  0.9542;  find  log  7. 

3.  Given  log  84  =  1.9243  and  log  12  =  1.0792;  find  log  7. 

4.  Given  log  1776  =  3.2494  and  log  1912  =  3.2815;  find  log 
1776/1912;  find  log  1912/1776. 

5.  Given  log  re/ 12  =  0.4321  and  log  12  =  1.0792,  find  log  x. 

139.  Logarithm  of  any  Power.  Let  n  be  any  positive  number 
and  let: 

logan  =  X  (1) 

From  (1),  by  the  definition  of  a  logarithm, 

71  =  a* 

Raising  both  sides  to  the  pth.  power,  where  p  is  any  number  what- 
soever, 

UP  =  ap* 

therefore,  by  definition  of  a  logarithm, 

logo(np)  =  px 
or  by  (1): 

loga(np)  =  p  log  all  (2) 

therefore  the  logarithm  of  any  power  of  a  number  equals  the  logarithm 
of  the  number  multiplied  by  the  index  of  the  power. 

The  above  includes  as  special  cases,  (1)  the  finding  of  the 
logarithm  of  any  integral  power  of  a  number,  since  in  this  case 
p  is  a  positive  integer,  or  (2)  the  finding  of  the  logarithm  of  any 
root  of  a  number,  since  in  this  case  p  is  the  reciprocal  of  the  index 
pf  the  root, 


§140]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   229 

Exercises 

1.  Given  log  2  =  0.3010;  find  log  1024;  find  log  V2;  find  logV2. 

2.  Given  log  1234  =  3.0913;  find  log  V1234.     Find  log  >7l234. 

3.  Given  log  5  =  0.6990;  find  log  5^^j^find  log  5^'^. 

4.  Simplify  the  expression  log  30/\/210  . 

Express  by  the  principles  established  in    §§137-139   the  following 
logarithms  in  as  simple  a  form  as  possible : 

5.  log  (i/9  ^  V3). 

6.  log  (^12  -^  V6). 

7.  log  {u^^  -i-  u^). 

8.  log  (10a26Va^^  b'). 

9.  Show  that  log  (11/15)  +  log  (490/297)  -  2  log  (7/9)  =  log  2. 
10.  Find  an  expression  for  the  value  of  x  from  the  equation  3*  =  567. 
Solution:     Take  the  logarithm  of  each  side 

a;  log  3  =  log  567 
But  log  567  =  log  (3*  X  7)  =  4  log  3  +  log  7 
therefore : 


X  log  3  =  4  log  3  +  log  7 


or: 


X  =  4  +  (log  7)/(log  3). 

11.  Find  an  expression  for  x  in  the  equation  5""  =  375. 

12.  Given  log  2  =  0.3010  and  log  3  =  0.4771,  find  how  many 
digits  in  6'°. 

13.  Find  an  expression  for  x  from  the  equation : 

3*  X  2^+1  =  V5I2. 

14.  Prove  that  log  (75/16)  -  2  log  (5/9)  +  log  (32/243)  =  log  2. 
140.  Characteristic   and   Mantissa.     The   common    logarithm 

of  a  number  is  always  written  so  that  it  consists  of  a  positive 
decimal  part  and  an  integral  part  which  may  be  either  positive 
or  negative.  Thus  log  0.02  =  log  2  -  log  100  =  0.3010  -  2. 
Log  0.02  is  never  written  —  1.6990. 

When  a  logarithm  of  a  number  is  thus  arranged,  special  names 
are  given  to  each  part.  The  positive  or  negative  integral  part  is 
called  the  characteristic  of  the  logarithm.  The  positive  decimal 
part  is  called  the  mantissa.  Thus,  in  log  200  =  2.3010,  2  is 
the  characteristic  and  3010  is  the  mantissa.  In  log  0.02  = 
0.3010  —  2,  (  —  2)  is  the  characteristic  and  3010  is  the  mantissa. 

Since  log  1  =  0  and  log  10  =  1,  every  number  lying  between  1 


230        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§140 

and  10  has  for  its  common  logarithm  a  proper  fraction — ^that 
is,  the  characteristic  is  0.  Thus  log  2  =  0.3010,  log  9.99  = 
0.9996,  log  1.91  ^  0.281.    Starting  with  the  equation: 

log  1.91  =  0.2810 
we  have,  by  §137, 

log  19.1  =  log  1.91  +  log  10  =  0.2810  +  1 
log  191  =  log  1.91  +  log  100  =  0.2810  +  2 
log  1910  =  log  1.91  +  log  1000  =  0.2810  +  3,  etc. 

Likewise,  by  §138, 

log  0.191  =  log  1.91  -  log  10  =  0.2810  -  1 
log  0.0191  =  log  1.91  -  log  100  =  0.2810  -  2 
log    0.00191  =  log  1.91  -  log  1000  =  0.2810  -  3,  etc. 

Since  the  characteristic  of  the  common  logarithm  of  any  number 
having  its  first  significant  figure  in  units  place  is  zero,  and  since 
moving  the  decimal  point  to  the  right  or  left  is  equivalent  to 
multiplying  or  dividing  by  a  power  of  10,  or  equivalent  to  adding 
an  integer  to  or  subtracting  an  integer  from  the  logarithm, 
(§135):  (1)  the  value  of  the  characteristic  is  dependent  merely 
upon  the  position  of  the  decimal  point  in  the  number;  (2)  the 
value  of  the  mantissa  is  the  same  for  the  logarithms  of  all 
numbers  that  differ  only  in  the  position  of  the  decimal  point. 
In  particular,  we  derive  therefrom  the  following  rule  for  finding 
the  characteristic  of  the  common  logarithm  of  any  number: 

The  characteristic  of  the  cornmon  logarithm  of  a  number  equals 
the  number  of  places  the  first  significant  figure  of  the  number  is 
removed  from  units'  place,  and  is  positive  if  the  first  significant 
figure  stands  to  the  left  of  units'  place  and  is  negative  if  it  stands 
to  the  right  of  units'  place. 

Thus  in  log  1910  =  3.2810,  the  first  figure  1  is  three  places  from 
units'  place  and  the  characteristic  is  3.  In  log  0.0191  =  0.2810 
—  2  the  first  significant  figure  1  is  two  places  to  the  right  of  units' 
place  and  the  characteristic  is  —  2.  A  computer  in  determining 
the  characteristic  of  the  logarithm  of  a  number  first  points  to 
units  place  and  counts  zero,  then  passes  to  the  next  place  and 
counts  one  and  so  on  until  the  first  significant  figure  is  reached. 

Logarithms    with    negative    characteristics,  like    0.3010  —  1, 


§141]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   231 

0.3010  —  2,  etc.,  are  frequently  written  in  the  equivalent  form 
9.3010  -  10,  8.3010  -  10,  etc. 

Exercises 

1.  What  numbers  have  0  for  the  characteristic  of  their  logarithm? 
What  numbers  have  0  for  the  mantissa  of  their  logarithms? 

2.  Find  the  characteristics  of  the  logarithms  of  the  following 
numbers:  1234,  5,678,910,  212,  57.45,  345.543,  7,  7.7,  0.7,  0.00000097, 
0.00010097. 

3.  Given  that  log  31,416  =  4.4971,  find  the  logarithms  of  the 
following  numbers:  314.16,  3.1416,  3,141,600,  0.031416,  0.31416, 
0.00031416. 

4.  Given  that  log  746  =  2.8727,  write  the  numbers  which  have  the 
following  logarithms:  4.8727,  1.8727,  0.8727  -  3,  0.8727  -  1,  3.8727, 
0.8727  -  4. 

141.  Logarithmic  Tables.  A  table  of  logarithms  usually  con- 
tains only  the  mantissas  of  the  logarithms  of  a  certain  con- 
venient sequence  of  numbers.  For  example,  a  four  place  table 
will  contain  the  mantissas  of  the  logarithms  of  numbers  from 
100  to  1000;  a  five  place  table  will  usually  contain  the  mantissas  of 
the  logarithms  of  numbers  from  1000  to  10,000,  and  so  on.  Of 
course  it  is  unnecessary  to  print  decimal  points  or  characteristics. 

A  table  of  logarithms  should  contain  means  for  readily  obtaining 
the  logarithms  of  numbers  intermediate  to  those  tabulated,  by 
means  of  tabular  differences  and  proportional  parts. 

The  tabular  differences  are  the  differences  between  successive 
mantissas.  If  any  tabular  difference  be  multiplied  successively 
by  the  numbers  0.1,  0.2,  0.3,  .  .  .  ,  0.8,  0.9,  the  results  are  called 
the  proportional  parts.  Thus,  from  a  four  place  table  we  find 
log  263  =  2.4200.  The  tabular  difference  is  given  in  the  table 
as  16.  If  we  wish  the  logarithm  of  263.7,  the  proportional 
part  0.7  X  16  or  11.2  is  added  to  the  mantissa,  giving,  to  four 
places,  log  263.7  =  2.4211.  This  process  is  known  as  interpola- 
tion. Corrections  of  this  kind  are  made  with  great  rapidity  after 
a  little  practice.  It  is  obvious  that  the  principle  used  in  the 
correction  is  the  equivalent  of  a  geometrical  assumption  that 
the  graph  of  the  function  is  nearly  straight  between  the  successive 
values  of  the  argument  given  in  the  table.     The  corrections 


232        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§142 

should  invariably  he  added  mentally  and  all  the  work  of  interpolation 
should  he  done  mentally  if  the  finding  of  the  proportional  parts 
hy  mental  work  does  not  require  multiplication  heyond  the  range  of 
12  X  12.  To  make  interpolations  mentally  is  an  essential  practice, 
if  the  student  is  to  learn  to  compute  by  logarithms  with  any  skill 
beyond  the  most  rudimentary  requirements. 

A  good  method  to  follow  is  as  follows:  Suppose  log  13.78  is 
required.  First  write  down  the  characteristic  1 ;  then,  with  the 
table  at  your  left,  find  137  in  the  number  column  and  mark  the 
corresponding  mantissa  by  placing  your  thumb  above  it  or  your 
first  finger  below  it.  Do  not  read  this  mantissa,  but  read  the 
tabular  difference,  32.  From  the  p.  p.  table  find  the  correction, 
26,  for  8.  Now  return  to  the  mantissa  marked  by  your  finger, 
and  read  it  increased  by  26,  i.e.,  1393;  then  place  1393  after 
the  characteristic  1  previously  written  down. 

The  accuracy  required  for  nearly  all  engineering  computations 
does  not  exceed  3  or  4  significant  figures.  Four  figure  accuracy 
means  that  the  errors  permitted  do  not  exceed  1  percent  of 
1  percent.  Only  a  small  portion  of  the  fundamental  data 
of  science  is  reliable  to  this  degree  of  accuracy.  ^  The  usual  meas- 
urements of  the  testing  laboratory  fall  far  short  of  it.  Only 
in  certain  work  in  geodesy,  and  in  a  few  other  special  fields  of 
engineering,  should  more  than  four  place  logarithms  be  used. 

142.  Anti-logarithms.  If  we  wish  to  find  the  number  which 
has  a  given  logarithm,  it  is  convenient  to  have  a  table  in  which 
the  logarithm  is  printed  before  the  number.  Such  a  table  is  known 
as  a  table  of  anti-logarithms.  It  is  usually  not  best  to  print 
tables  of  anti-logarithms  to  more  than  four  places;  to  find  a  number 
when  a  five  place  logarithm  is  given,  it  is  preferable  to  use  the 
table  of  logarithms  inversely,  as  the  large  number  of  pages  required 
for  a  table  of  anti-logarithms  is  a  disadvantage  that  is  not  com- 
pensated for  by  the  additional  convenience  of  such  a  table. 

1  Fundamental  constants  upon  which  much  of  the  calculation  in  applied 
science  must  be  based  are  not  often  known  to  four  figures.  The  mechanical 
equivalent  of  heat  is  hardly  known  to  1  percent.  The  specific  heat  of  super- 
heated steam  is  even  less  accurately  known.  The  tensile,  tortional  and  com- 
pressive strength  of  no  structural  material  would  be  assumed  to  be  known  to  a 
greater  accuracy  than  the  above-named  constants.  Of  course  no  calculated 
result  can  be  more  accurate  than  the  least  accurate  of  the  measurements  upon 
which  it  depends. 


§1431  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS    233 

143.  Cologarithms.  Any  computation  involving  multiplica- 
tion, division,  evolution  and  involution  may  be  performed  by 
the  addition  of  a  single  column  of  logarithms.  This  possibility 
is  secured  by  using  the  cologarithm,  instead  of  the  logarithm,  of 
all  divisors.  The  cologarithm,  or  complementary  logarithm, 
of  a  number  n  is  defined  to  be  (10  —  log  n)  —  10.  The  part 
(10  —  log  n)  can  be  taken  from  the  table  just  as  readily  as  log  n, 
by  subtracting  in  order  all  the  figures  of  the  logarithm,  including  the 
characteristic,  from  9,  except  the  last  figure,  which  must  be  taken 
from  10.  The  subtraction  should,  of  course,  be  done  mentally. 
Thus  log  263  =  2.4200,  whence  colog  263  =  7.5800  -  10.  It 
is  obvious  that  the  addition  of  (10  —  log  n)  —  10  is  the  same 
as  the  subtraction  of  log  n. 

The  convenience  arising  from  this  use  may  be  illustrated  as 
follows: 
Suppose  it  is  required  to  find  x  from  the  proportion 


37.42  :x::Q4:7  :    Vo.582. 

We  then  have 

2  log  37.4  =  3.1458 
(1/2)  log  0.582  =  9.8825  -  10 
colog  647  =  7.1891  -  10 
log  [1.650]  =  0.2174 

Therefore  x  =  1.650. 

It  is  a  good  custom  to  enclose  a  computed  result  in  square 
brackets. 

144.  Arrangement  of  Work.  All  logarithmic  work  should  be 
arranged  in  a  vertical  column  and  should  be  done  with  pen  and 
ink.  Study  the  formula  in  which  numerical  values  are  to  be 
substituted  and  decide  upon  an  arrangement  of  your  work  in  the 
vertical  column  which  will  make  the  additions,  subtractions,  etc., 
of  logarithms  as  systematic  and  easy  as  possible.  Fill  out  the 
vertical  column  with  the  names  and  values  of  the  data  before 
turning  to  the  table  of  logarithms.  This  is  called  blocking  out 
the  work.  The  work  is  not  properly  blocked  out  unless  every 
entry  in  the  work  as  laid  out  is  carefully  labelled,  stating  exactly 
the  name  and  value  of  the  magnitude  whose  logarithm  is  taken. 


234        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§144 

and  unless  the  computation  sheet  bears  a  formula  or  statement 
fully  explaining  the  purpose  of  the  work. 

Computation  Sheet,  Form  ilf  7,  is  suitable  for  general  logarithmic 
computation. 

Exercises 

1.  From  a  four  place  table  find  the  logarithms  of  the  following 
numbers:  342,  1322,  8000,  872.4,  35.21,  0.00213,  3.301,  325.67, 
21,  3.1416,  0.0186,  250.75,  0.0007,  0.33333. 

2.  Find  the  numbers  corresponding  to  each  of  the  following 
logarithms:  0.3250,  2.1860,  0.8724,  1.1325,  3.0075,  8.3990  -  10, 
9.7481  -  10,  4.0831,  7.0091  -  10,  0.5642. 

3.  Compute  by  logarithms  the  value  of  the  following:  2.56  X  3.11 
X  421;  7.04  X  0.21  X  0.0646;  3215  X  12.82  --  864.  

4.  Compute  the  following  by  logarithms:  81^  -^  17^;  158'Nyo.52; 
(343/892)3;   Vl893  Vl912/4462. 

5.  Compute  the  following  by  logarithms:   (2.7182)i-40"';  (7.41)-'/  ; 

(8.31)0-27. 

6.  Solve  the  following  equations:  5^  =  10;  3*-i  =  4;  logx  71  =  1.21 
log.  5  =  logio  4.822. 

7.  Find  the  amount  of  $550  in  fifteen  years  at  5  percent  com- 
pound interest. 

8.  A  corporation  is  to  repay  a  loan  of  $200,000  by  twenty  equal 
annual  payments.  How  much  will  have  to  be  paid  each  year,  if 
money  be  supposed  to  be  worth  5  percent? 

Let  X  be  the  amount  paid  each  year.  As  the  debt  of  $200,000  is 
owed  now,  the  'present  value  of  the  twenty  equal  payments  of  x  dollars 
each  must  add  up  to  the  debt  or  $200,000.  The  sum  of  x  dollars 
to  be  paid  n  years  hence  has  a  present  worth  of  only 

X 

(1.05)« 
if  money  be  worth  5  percent  compound  interest.     The  present  value, 
then,  of  X  dollars  paid  one  year  hence,  x  dollars  paid  two  years  hence, 
and  so  on,  is 

1.05  ^  (1.05)2  ^  (1.05)3  -r    •    •    •     t-  (1.05)20 

This  is  a  geometrical  progression. 

The  result  in  this  case  is  the  value  of  an  annuity  payable  at  the 
end  of  each  year  for  twenty  years  that  a  present  payment  of  $200,000 
will  purchase. 


§145]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   235 

9.  It  is  estimated  that  a  certain  power  plant  costing  $220,000  will 
become  entirely  worthless  except  for  a  scrap  value  of  $20,000  at  the 
end  of  twenty  years.  What  annual  sum  must  be  set  aside  to  amount 
to  the  cost  of  replacement  at  the  end  of  twenty  years,  if  5  percent 
compound  interest  is  realized  on  the  money  in  the  depreciation 
fund? 

Let  the  annual  amount  set  aside  be  x.  In  this  case  the  twenty 
equal  payments  are  to  have  a  value  of  $200,000  twenty  years  hence, 
while  in  the  preceding  problem  the  payments  were  to  be  worth 
$200,000  now.     In  this  case,  therefore, 

x(1.05)^9  +  a;(1.05)i8  +  aiCLOS)!"'  +  .    .    . 

+x(1.05)2  +  x(1.05)  +x  =  $200,000. 

The  geometrical  progression  is  to  be  summed  and  the    resulting 
equation  solved  for  x. 

10.  The  population  of  the  United  States  in  1790  was  3,930,000  and 
in  1910  it  was  93,400,000.  What  was  the  average  rate  percent  in- 
crease for  each  decade  of  this  period,  assuming  that  the  population 
increased  in  geometrical  progression  with  a  uniform  ratio  for  the  entire 
period. 

11.  Find  the  surface  and  the  volume  of  a  sphere  whose  radius  is 
7.211. 

12.  Find  the  weight  of  a  cone  of  altitude  9.64  inches,  the  radius 
of  the  base  being  5.35  inches,  if  the  cone  is  made  of  steel  of  specific 
gravity  7.93. 

13.  Find  the  weight  of  a  sphere  of  cast  iron  14.2  inches  in  diameter, 
if  the  specific  gravity  of  the  iron  be  7.30. 

14.  In  twenty-four  hours  of  continuous  pumping,  a  pump  discharges 
450  gallons  per  minute;  by  how  much  will  it  raise  the  level  of  water  in 
a  reservoir  having  a  surface  of  1  acre?     (1  acre  =  43560  sq.  ft.) 

145.  Trigonometric  Computations.  Logarithms  of  the  trig- 
onometric functions  are  used  for  computing  the  numerical  value 
of  expressions  containing  trigonometric  functions,  and  in  the 
solution  of  triangles.  The  right  triangles  previously  solved  by 
use  of  the  natural  functions  are  often  more  readily  solved  by 
means  of  logarithms.  (See  §66.)  The  tables  of  logarithmic  func- 
tions contain  adequate  explanation  of  their  use,  so  that  de- 
tailed instructions  need  not  be  given  in  this  place.  Two  new 
matters  of  great  importance  are  met  with  in  the  use  of  the  loga- 
rithms of  the  trigonometric  functions  that  do  not  arise  in  the  use 


236        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§145 

of  a  table  of  logarithms  of  numbers,  which,  on  that  account,  require 
especial  attention  from  the  student: 

(1)  In  interpolating  in  a  table  of  logarithms  of  trigonometric 
functions,  the  corrections  to  the  logarithms  of  all  co-functions  must 
be  subtracted  and  not  added.  Failure  to  do  this  is  the  cause  of 
most  of  the  errors  made  by  the  beginner. 

(2)  To  secure  proper  relative  accuracy  in  computation,  the 
S  and  T  functions  must  be  used  in  interpolating  for  the  sine  and 
tangent  of  small  angles. 

In  the  following  work,  four  place  tables  of  logarithms  are 
supposed  to  be  in  the  hands  of  the  students. 


Exercises 

1.  A  right  prism,  whose  base  is  a  square  17.45  feet  on  a  side,  is 
cut  by  a  plane  making  an  angle  of  27°  15'  with  a  face  of  the  prism. 
Find  the  area  of  the  section  of  the  prism  made  by  the  cutting  plane. 

2.  The  perimeter  of  a  regular  decagon  is  24  feet.  Find  the  area  of 
the  decagon. 

3.  To  find  the  distance  between  two  points  B  and  C  on  opposite 
banks  of  a  river,  a  distance  CA  is  measured  300  feet,  perpendicular 
to  CB.  At  A  the  angle  CAB  is  found  to  be  47°  27'.  Find  the 
distance  CB. 

4.  In  running  a  line  18  miles  in  a  direction  north,  2°  13.2'  east, 
how  far  in  feet  does  one  depart  from  a  north  and  south  line  passing 
through  the  place  of  beginning? 

5.  How  far  is  Madison,  Wisconsin,  latitude  43°  5',  from  the  earth's 
axis  of  rotation,  assuming  that  the  earth  is  a  sphere  of  radius  3960 
miles? 

6.  Find  the  length  of  the  belt  required  to  connect  an  8-foot  and  a 
3-foot  pulley,  their  axes  being  21  feet  apart. 

7.  A  man  walking  east  7°  15'  north  along  a  river  notices  that  after 
passing  opposite  a  tree  across  the  river  he  walks  107  paces  before  he 
is  in  line  with  the  shadow  of  the  tree.  Time  of  day,  noon.  How  far 
is  it  across  the  river? 

8.  Solve  the  right-angled  triangle  in  which  one  leg  =  2  \/S  and  the 
hypotenuse  =  27r. 

9.  The  moon's  radius  is  1081  miles.  When  nearest  the  earth,  the 
moon's  apparent  diameter  (the  angle  subtended  by  the  moon's  disk  as 
seen  from  the  position  of  the  earth's  center)  is  32.79'.     When  farthest 


§146]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   237 

from  the  earth,  her    apparent  diameter    is  only  28.73'.     Find  the 
nearest  and  farthest  distances  of  the  moon  in  miles. 

10.  A  pendulum  39  inches  long  vibrates  3°  5'  each  side  of  its  mean 
position.  At  the  end  of  each  swing,  how  far  is  the  pendulum  bob 
above  its  lowest  position? 

11.  If  the  deviation  of  the  compass  be  2°  1.14'  east,  how  many  feet 
does  magnetic  north  depart  from  true  north  in  a  distance  of  1  mile 
true  north? 

12.  Solve:  _ 

X  :  1.72  =  427  :  ^y'2gh 

a  g  =  32.2  and  h  =  78.2. 

13.  The  four  strings  of  a  violin  are  tuned  in  fifths;  that  is,  for  two 
vibrations  of  any  string  there  are  three  vibrations  of  the  next  higher 
string.  If  the  lowest  or  G  string  vibrates  196  times  per  second,  find 
the  number  of  vibrations  per  second  of  the  highest  string. 

14.  A  substance  containing  20  percent  of  impurities  is  to  be  purified 
by  crystallization  from  a  mother  liquid.  Each  crystallization  reduces 
the  impurity  88.6  percent.  How  many  crystallizations  will  produce 
a  substance  0.9999  pure? 

2ir 

15.  Compute  the  value  of  (1  —  ae^^Y  where  a  =  15.6,  h  =  ..  ' 

X  =  10,  n  =  2,  2/  =  2.5. 

16.  Find  the  volume  of  a  cone  if  the  angle  at  the  apex  be  15°  38' 
and  the  altitude  17.48  inches. 

17.  The  angle  subtended  by  the  sun's  diameter  as  seen  from  the 
earth  is  32'. 06.  Find  the  diameter  of  the  sun  in  miles,  if  the  distance 
from  the  earth  to  the  sun  be  92,8  million  miles. 

18.  Compute  by  logarithms  four  values  of  p  from  the  equation 
p  =  32.2di"8,  for  d  =  2,  3,  4,  5. 

19.  Solve  3*  =  405  for  the  value  of  x. 

20.  Compute: 

23.07  X  0.1354  X  V234. 
13.54 
What  advantage  is  there  in  using  the  co-logarithm  of  the  denomi- 
nator? 

146.  Logarithmic  and  Exponential  Curves.  The  graphical 
construction  of  the  exponential  curve  has  already  been  explained. 
It  was  noted  that  curves  whose  equations  are  of  the  form  y  =  r' 
pass  through  the  point  (0,  1)  and  that  the  slope  of  the  curves 
for  positive  values  of  a?  is  steeper  the  larger  ^ihe  value  selected  for 


238        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§146 

the  number  r.  See  Fig.  94.  In  a  system  of  exponential  curves 
y  =  r'  passing  through  the  point  (0,  1)  or  the  point  M  of  Fig. 
94,  we  shall  assume  that  there  is  one  curve  passing  through  M 
with  slope  1.  The  equation  of  this  particular  curve  we  shall  call 
y  =  e",  thereby  defining  the  number  e  as  that  value  of  r  for  which 
the  curve  y  =  r''  passes  through  the  point  (0,  1)  with  slope  1.  This 
is  a  second  definition  of  the  number  e;  we  shall  show  in  this  section 
that  it  is  consistent  with  the  first  definition  of  e  given  in  §130. 


Fig.  94. — Definition  of  Tangent  to  a  Curve. 


The  exercises  of  §130  developed  experimentally  the  charac- 
teristic property  of  the  exponential  curve  to  the  base  e: 
The  slope  of  the  curve  y  =  e""  at  any  point  is  equal  to  the  ordinate 
of  that  point.  This  fact,  developed  experimentally  in  §130, 
will  now  be  shown  to  follow  necessarily  from  the  definition  of  e 
just  given. 

Select  the  point  P  on  the  curve  ?/  =  e*  at  any  point  desired. 

Draw  a  line  through  P  cutting  the  curve  at  any  neighboring 
point  Q.  (Fig.  94.)  A  line  like  PQ  that  cuts  a  curve  at  two  points 
is  called  a  secant  line.  As  the  point  Q  is  taken  nearer  and  nearer 
to  the  point  P  (P  r^njaining  fixed),  the  limiting  position  ap- 


§146]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   239 

preached  by  tKe  secant  PQ  is  called  the  tangent  to  the  curve  at 
the  point  P.  This  is  the  general  definition  of  the  tangent  to  any 
curve. 

The  slope  of  the  secant  joining  P  to  the  neighboring  point  Q 
is  HQ/PH.  As  the  point  Q  approaches  P  this  ratio  approaches 
the  slope  of  the  tangent  to  y  =  6"=  at  the  point  P.  Let  OD 
=  X  and  PH  =  h;  then  OE  =  x  -\-  h,  also  DP  =  e*  and  EQ  = 
6*+'^.  Since  HQ  is  the  y  of  the  point  Q  minus  the  y  of  the 
point  P,  we  have: 

HQ  _  e^+^-e^  _     e^-1 

PH~        h        ~  ^'    h 

Now  the  slope  of  2/  =  e^  at  P  is  the  limit  of  the  above  expression 

as  Q  approaches  P  or  as  /i  approaches  zero.     That  is: 

limit  e''  -1  ,,. 

slope  of  e^  at  P  =  e^,  ^  „ — r —  (I) 

We  now  seek  to  find 

limit  e^  —  1 

h  =  0      h 
if  such  limit  exists.     Since  P  is  any  point,  consider  the  point  M 
where  x  =  0.     The  slope  there  is: 

„  limit  e''  —  1 
A  =  0     h 
That  is,  the  slope  of  2/  =  e*  at  M  is: 

limit  e^  —  1 

h  =  G      ~h 

But  by  the  definition  of  e,  the  slope  of  ?/  =  e*  at  Af  is  L  Hence 
we  must  conclude  that  the  required  limit  exists  and  that 

limH     e'^-l_.  /^x 

h  =  o      h     -  ^  ^^^ 

Substituting  this  result  in  equation  (1),  we  have 

Slope  at  P  =  e*  (3) 

This  expresses  the  fact  that  the  slope  oi  y  =  e'  at  any  point  is  e*, 
or  is  the  ordinate  y  of  that  point,  a  fact  that  was  first  indicated 
experimentally  in  §130.  At  that  same  place  the  approxi- 
mate value  of  e  was  seen  to  be  2.7.  A  more  exact  value  is  known 
to  be  2.7183,  as  will  be  computed  later. 


240        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§146 


In  Fig.  94  the  slope  of  ?/  =  e^  at  P  is  given  by  PD  measured  by 
the  unit  OM.  The  distance  TD,  called  the  subtangent,  is 
constant  for  all  positions  of  the  point  P. 

The  slope  of  y  =  r''  at  any  point  is  readily  found.  There 
exists  a  number  m  such  that  e"*  =  r.  Hence  y  =  r""  may  be 
written  y  =  (e"»)*  =  e'"^.  Now  this  curve  is  made  from  2/  =  e^ 
by  substituting  mx  for  x,  or  by  multiplying  all  of  the  abscissas 
of  the  latter  by  1  jm.  Therefore  the  side  TD  of  the  triangle  PDT 
in  Fig. '94  will  be  multiplied  by  1  /m,  the  other  side  DP  remaining 


19 

a 

18 

17 

;* 

H 

h 

_ll_ 
H 

9) 

15 

14 

0 

H 

n 

IS 

« 

^ 

1? 

11 

10 

9 

7 

fi 

fi 

4 

-3 

M 

=  1 

ogeajj 



-^- 

- 

l\ 

^ 

-— 

— ' 

x:=e2^ 

^ 

■ 

-4 

-3 

-2 

-1 

-1 

3 

4 

5 

6 

7 

8 

9 

10 

^12 

13i4| 

-9 

"~~ 



J^=loge«| 

-S 

~ 

■+- 

_ 

-4 

a- 

e" 

y 

-5 

Fig.  95. — Exponential  and  Logarithmic  Curves  to  the  Natural  Base  e  = 

2.7183. 

the  same.  Therefore  the  slope  of  the  curve,  or  DP/TD  will  be 
multiplied  by  m,  since  the  denominator  of  this  fraction  is  multi- 
plied by  1  jm.  Hence  the  slope  oi  y  —  r""  at  any  point  is  m  times 
the  ordinate  of  that  point,  where  m  satisfies  the  equation  e'"  =  r. 

The  curve  y  =  e~*  is,  of  course,  the  curve  y  =  e""  reflected  in 
the  F-axis.^  This  curve,  as  well  as  the  curve  y  =  loge  x  and  its 
symmetrical  curve,  are  shown  in  Fig.  95.  Sometimes  the  curve 
2/  =  e^  is  called  the  exponential  curve  and  the  curve  y  =  loge  x 
is  called  the  logarithmic  ciirve.     This  distinction,  however,  has 

'  See  §24. 


§146]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   241 

little  utility,  as  the  equation  of  either  locus  can  be  expressed  in 
either  notation. 

The  notation  y  =  In  x  is  often  used  to  indicate  the  natural  loga- 
rithm of  X  and  the  notation  y  =  Ig  x  or  y  =  log  x  is  used  to  stand 
for  the  common  logarithm  of  x. 

TABLE  IV. 

The  following  table  of  powers  of  e  is  useful  in  sketching  exponential 
curves. 


e 


=  1.2214 
=  1.4918 
=  1.8221 
=  2.2255 
=  2.7183 
=  7.3891 
=  20.0855 
=  54.5982 


e>2  =  1.6487 
e'^  =  1.3956 
e^  =  1.2840 
e'/°  =  1.2214 


=  0.8187 
=  0.6703 
=  0.5488 
=  0.4493 
=  0.3679 
=  0.1353 
=  0.0498 
=  0.0183 


\ 

\ 

u 

o 

\1 

/ 

V 

/ 

V 

w»  0 

m  =  0 

■^^;0^ 

^^ 

-2-10  12 

Fig.  96. — A  Family  of  Exponentials,  y  =  V 


Exercises 

L  Draw  the  curve  y  =  g'  -\-  e~*.     Show  that  y  is  an  even  function 
of  X,  that  is,  that  y  does  not  change  when  the  sign  of  x  is  changed. 
16 


242        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§147 

2.  Draw  the  curve  y  =  e^  —  e~*.  Show  that  this  is  an  odd  func- 
tion of  X,  that  is,  that  the  function  changes  sign  but  not  absolute  value 
when  the  sign  of  x  is  changed. 

3.  Draw  the  graphs  oi  y  =  e^^^,  and  y  =  e*/^. 

4.  Draw  the  graphs  oi  y  =  e''^^,  and  y  =  e~'^^. 

5.  Compare  the  curves:  y  =  e""^^,  y  =  e^^"^,  y  =  e",  y  =  e'*. 

6.  Sketch  the  curves  7/  =  1^,  2/  =  2*,  2/  =  3*,  y  =  ^'',  y  =  5^, 
?/  =  6^,  2/  =  8^,  2/  =  10^,  from  x  =  -3tox  =  +3. 

7.  From  the  graphs  of 

y  =  x"^ 
and 

y  =  log  lox  +  1.8 
solve  the  equation 

x"^  -  log  X  -  1.8  =  0. 

8.  Solve  graphically  the  equation 

5  log  X  -  (l/2)x  +  2=0. 

9.  Solve  graphically : 

10^  =  x\ 

10.  Solve  graphically: 

(1/2)-  =  log  X. 

11.  Solve  graphically: 

\Qx  =  s-sin  X. 

12.  Solve  graphically : 

sin  X  =  X  —  Q.\. 

13.  Solve  graphically: 

cos  X  =  x^  —  \. 

14.  Solve  analytically : 

147.  The  Exponential  Curve  and  the  Theorems  on  Loci.     It  has 

already  been  shown  (§  145)  that  the  curve  ?/  =  a-  can  be  derived 
from  the  curve  y  =  e""  (a>e)  by  multiplying  the  abscissas  of 
the  latter  curve  by  1  /m  (m  >  1),  that  is,  by  orthographic  projection 
of  ?/  =  e-  upon  a  plane  passing  through  the  F-axis.  There  exists 
a  number  m  (m>l)  such  that  a  =  e'^.  Hence,  y  =  a"  may  be 
written  y  =  e'"*  and,  by  §27,  the  latter  curve  may  be  made 
from  1/  =  e*  by  multiplying  its  abscissas  by  1  /m.  Also  note  that 
the  slope  of  the  curve  ?/  =  e*  at  any  point  is  equal  to  the  ordinate 
of  the  point,  and  that  the  slope  oi  y  =  a"  at  any  point  is  m 
times  the  ordinate  of  that  point.  The  number  1/m  is  called  the 
modulus  of  the  logarithmic  system  whose  base  is  a. 


(1) 

(2) 


§147]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   243 

The  modulus  of  the  common  syatem  is  the  reciprocal  of  the  value 
of  m  that  satisfies  e""  =  10,  or  it  is  the  value  of  M  that  satisfies 
gi/M  =  10,  or  that  satisfies  e  =  10^.  That  is,  the  modulus  M  of 
the  common  system  is  the  logarithm  of  e  to  the  base  10,  or,  to  four 
figures,  equals  0.4343.  The  value  of  m  or  1  /M  =  2.3026.  Thus 
we  have  the  fundamental  formulas: 

100-4343   _    g| 
e2.3026   =    10) 

and 

logioA^  =  0.4343  logeiV  I 
logeN  =  2.3026  logioiV) 

Another  remarkable  property  of  the  logarithmic  curve  ap- 
pears from  comparing  the  curves  y  =  a''  and  y  =  a*+\  or,  more 
generally,  the  curves  y  =  a""  and  y  =  a"'^^.  The  second  of  these 
curves  can  be  derived  from  y  =  a^'hj  translating  the  latter  curve 
the  distance  1  (in  the  general  case  the  distance  h)  to  the  left. 
But  y  =  a^+^  may  be  written  y  =  aa'',  and  y  =  a^+^  may  be 
written  y  =  a^a"".  From  these  it  can  be  seen  that  the  new  curves 
may  also  be  considered  as  derived  from  i/  =  a^  by  multiplying  all 
ordinates  oi  y  =  a''  by  a,  or  in  the  general  case,  by  a^. 

Translating  the  exponential  curve  in  the  x-direction  is  the  same  as 
multiplying  all  ordinates  by  a  certain  fixed  number,  or  is  equivalent 
to  a  certain  orthographic  projection  of  the  original  curve  upon  a  plane 
through  the  X-axis. 

Changing  the  sign  of  h  changes  the  sense  of  the  translation  and 
changes  elongation  to  shortening  or  vice  versa. 

The  exponential  curve  might  be  defined  as  the  locus  that 
possesses  the  above-described  fundamental  property.  There  are 
numerous  ways  in  which  this  property  may  be  stated.  Another 
form  is  this:  Any  portion  of  the  exponential  curve  included  within 
any  interval  of  x,  may  be  made  from  the  portion  of  the  curve 
included  within  any  other  equal  interval  of  x,  by  the  elongation 
(or  shortening)  of  the  ordinates  in  a  certain  ratio,  or,  in  other 
words,  by  orthographic  projection  upon  a  plane  passing  through 
the  x-Sixis.  This  is  illustrated  by  Fig.  97,  which  is  a  graph  of  an 
exponential  curve  drawn  to  base  2.  If  the  portions  of  the  curve 
PiPi,  PiPz,  PzPa,   .    .    .   corresponding  to  equal  intervals  1  of  x 


244        ELEMENTARY  MATHEMATICAL  ANALYSIS 


J147 


be  changed  by  shortening  all  ordinates  of  P1P2  measured  above  the 
height  of  Pi  in  the  ratio  1  /2,  by  shortening  all  ordinates  of  P2P3 
measured  above  P2  in  the  ratio  1  /4,  by  shortening  all  ordinates  of 
PgP4  measured  above  P3  in  the  ratio  1  /8,  .  .  .  the  results  are 
the  curves  PiPi,  P2P2,  P3P3,  •  •  •  which  are  identical  with  the 
portion  PyPi  of  the  original  curve. 


9 

8 

" 

1/ 

F, 

7 

Pz 

6 

5 

4 

j  / 

F, 

3 

/ 

Pi 

2 

/'''" 

Fi 

/ 

^,^ 

P] 

1 

1 

-2-10  12  345 

Fig.  97. — Illustration    of   an    Important    Property    of    the    Exponential 

Curve. 


This  is  also  illustrated  by  Fig.  93,  which  is  a  small  portion  of  the 
curve  X  =  log  10  y  drawn  on  a  large  scale,  and,  for  convenience,  with 
the  vertical  unit  I  /lO  the  horizontal  unit.  From  this  small  portion 
of  the  curve  we  may  read  the  logarithms  of  all  numbers.  For  the 
distances  along  the  a;-axis  may  be  designated  0.0,  0.1,  0.2,  .  .  . 
or  1.0,  I..1,  1.2,  ...  or  2.0,  2.1,  2.2,  .  .  .,  etc.,  in  which  case 
weread  1,2,3,  .  .  .  or  10, 20,30,  .  .  .  or  100, 200, 300,  .  .  . 
etc.,  respectively,  along  the  y-axis.  This,  it  will  be  observed,  is 
merely  a  geometrical  statement  of  the  fact  that  a  table  of  man- 
tissas for  the  numbers  from  1.000  to  9.999  is  sufficient  for  deter- 
mining the  logarithms  of  all  four-figure  numbers. 


§148]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   245 

Exercises 

1.  State  the  difference  between  the  curves  ?/  =  e^  and  y  =  10*. 

2.  Graph  y  =  e"""  where  e  =  2.7183. 

3.  Graph  the  logarithmic  spiral  p  =  e\    d  being   measured    in 
radians. 

Note  :  The  radian  measure  in  the  margin  of  Form  M3  should  be 
used  for  this  purpose. 

4.  Graph  p  =  e~^ 

5.  The  pressure  of  the  atmosphere  is  given  in  millimeters  of  mer- 
cury by  the  formula: 

y  =  760 -e-^"  8000 

where  the  altitude  x  is  measured  in  meters  above  the  sea  level.  Pro- 
duce a  table  of  pressure  for  the  altitudes  x  =  0;  10;  50;  100;  200;  300; 
1000;  10,000;  100,000. 

6.  From  the  data  of  the  last  problem,  find  the  pressure  at  an  alti- 
tude of  25,000  feet. 

7.  Show  that  the  relation  of  Exercise  5  may  be  written : 

X  =  18,421  (log  760  -  log  y). 

8.  Determine  the  value  of  the  quotient  ^j —  for  the  following 

values  of  x:  2,  3,  5,  7. 

9.  How  large  is  e°°°^  approximately? 

10.  What  is  the  approximate  value  of  10°°°^? 

148.  Logarithmic  Double  Scale.  The  relation  between  a  num- 
ber and  its  logarithm  can  be  shown  by  a  double  scale  of  the  sort 
discussed  in  §§3  and  8.  In  constructing  the  double  scale, 
one  may  select  for  the  uniform  scale  either  the  one  on  which  the 
numbers  are  to  be  read,  or  the  one  on  which  the  logarithms  are  to 
be  read.  A  scale  having  a  most  remarkable  and  useful  property 
results  if  the  logarithms  are  laid  off  on  a  uniform  scale  and  the 
corresponding  numbers  are  laid  off  on  a  non-uniform  scale,  as 
shown  in  the  double  scale  of  Fig.  98.  This  scale  is  constructed 
for  the  base  10.  The  distances  measured  on  the  5-scale,  although 
it  is  the  scale  on  which  the  numbers  are  read,  are  proportional  to 
the  common  logarithms  of  the  successive  numbers;  that  is,  if  the 
total  length  of  the  scale  be  called  unity,  the  distance  on  the  B 
scale  from  the  left  end  to  the  mark  2  is  0.3010,  the  distance  to  the 
mark  3  is  0.4771,  etc.;  also  the  distance  on  this  scale  from  the  left 
end  to  the  mark  6  is  the  sum  of  the  distance  from  the  left  end  to 


246        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§148 


the  mark  2  and  the  like  distance  to  the  mark  3;  also  the  distance 
to  8  is  just  treble  the  distance  to  2. 
Since  log  IOj;  =  1  +  log  x,  it  follows  that,  if  the  scales  A  and  B, 
Fig.  98,  were  extended  another  unit  to  the  right, 
this  second  unit  would  be  identical  to  the  first 
one,  except  in  the  attached  numbers.  The 
numbers  on  the  ^-scale  would  be  changed  from 
0.0,  0.1,  0.2,  .  .  .  LO  to  1.0,  1.1,  1.2,  .  .  ., 
2.0,  while  those  on  the  non-uniform,  or  5-scale, 
would  be  changed  from  1,  2,  3,  .  .  . ,  10  to 
10,  20,  30,   .    .    .    100. 

Passing  along  this  scale  an  integral  number 
of  unit  intervals  corresponds  thus  to  chiange  of 
characteristic  in  the  logarithms,  or  to  change 
of  decimal  point  in  the  numbers. 

It  is  not,  however,  necessary  to  construct 
more  than  one  block  of  this  double  scale,  since 
we  are  at  liberty  to  add  an  integer  n  to  the 
numbers  of  the  uniform  scale,  provided  at  the 
same  time  we  multiply  the  numbers  of  the 
non-uniform  scale  by  10".  In  this  way  we 
may  obtain  any  desired  portion  of  the  extended 
scale.  Thus,  we  may  change  0.1,  0.2, 0.3,  .  .  ., 
1.0  on  A  to  3.1,  3.2,  3.3,  .  .  .,  4.0,  by  adding 
3  to  each  number,  provided  at  the  same  time 
we  change  the  numbers  on  the  -B-scale  1,  2,  3,  • 
4,  .  .  .,  10  to  1000,  2000,  3000,  4000,  .  .  . 
10,000  by  multiplying  them  by  10^.  If  n  is 
negative  (say  —  2)  we  may  write,  as  in  the 
case  of  logarithms,  8.0  -  10,  8.1  -  10,  8.2  - 
10,  .  .  . ,  9.0  —  10,  or,  more  simply,  —  2,  — 
1.9,  -  1.8,  -  1.7,  .  .  .,  -  1.0,  changing  the 
numbers  on  the  non-uniform  scale  at  the  same 
time  to  0.01,  0.02,  0.03,   .    .    .,0.10. 

To  produce  the  scale  of  distances  proportional  to  the  logarithms  ' 
of  the  successive  numbers  as  used  above,  it  is  merely  necessary  to 
draw  horizontal  lines  through  the  points  1,  2,  3,   .    .    .of  the 
2/-axis  in  Fig.  99,  and  then  draw  vertical  lines  through  the  points 


%  - 

"a   N 
w   _: 


=  E  ..       S 


,1 


-=  ^ 


§149]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   247 

P2,  P3,  Pi  .    .    .   where  the  horizontal  lines  meet  the  curve;  the 
intercepts  on  the  a;-axis  are  then  proportional  to  log  x. 

149.  The  Slide  Rule.  By  far  the  most  important  application 
of  the  non-uniform  scale  ruled  proportionally  to  log  x,  is  the  com- 
puting device  known  as  the  slide  rule.  The  principle  upon  which 
the  operation  of  the  slide  rule  is  based  is  very  simple.     If  we  have 


N 


10 

1001000 

^J 

^w 

9 

90 

9 

^^ 

/ 

', 

8 

r 

/ 

/ 

80 

8 

/ 

'^7 

k 

r 

/ 

Hs 

J 

7 

70 

7 

0 

/] 

'h, 

/ 

/ 

/ 

p-l 

6 

60 

6( 

0 

A 

H 

4 

/ 

/ 

/ 

Pf, 

5 

50 

5( 

0 

/ 

{H, 

/ 

/ 

/ 

Pb 

4 

10 

4 

)0 

^ 

/ 

/u 

/ 

Pi 

3 

30 

3 

/ 

/ 

/ 

y 

k 

2 

20 

^2 

^ 

y^ 

y 

^ 

^ 

I' 

i 

iil 

-t 

)0 

2 

3 

4 

5 

p 

8 

91) 

6 

0 

tFo 

U 

4 

0 

C 

08 

10 

L 

u 

2l2 

u 

J 

Lj 

Li 

IL 

i8_ 

LJ 

J 

J 

/  15   % 


Fig.  99. — A  Method  of  Constructing  the  Logarithmic  Scale. 


two  scales  divided  proportionally  to  log  x  {A  and  B,  Fig.  100), 
so  arranged  that  one  scale  may  slide  along  the  other,  then  by  slid- 
ing one  scale  (called  the  slide)  until  its  left  end  is  opposite  any 
desired  division  of  the  first  scale,  and,  selecting  any  desired  division 
of  the  slide,  as  at  R,  Fig.  100,  taking  the  reading  of  the  original 
scale  beneath  this  point,  as  N,  the  product  of  the  two  factors 
whose  logarithms  are  proportional  to  AB  and  BR  can  be  read 
directly  from  the  lower  scale  at  N;  for  AN  is,  by  construction, 
the  sum  of  AB  and  BR,  and  since  the  scales  were  laid  off  propor- 
tionally to  log  X,  and  marked  with  the  numbers  of  which  the  dis- 
tances are  the  logarithms,  the  process  described  adds  the  logarithms 
mechanically,  but  indicates  the  results  in  terms  of  the  numbers 


248        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§149 

themselves.  By  this  device  all  of  the  operations  commonly  carried 
out  by  use  of  a  logarithmic  table  may  be  performed  mechanically. 
Full  description  of  the  use  of  the  slide  rule 
need  not  be  given  in  detail  at  this  place,  as 
complete  instructions  are  found  in  the  pamph- 
lets furnished  with  each  slide  rule.  A  very 
brief  amount  of  individual  instruction  given  to 
the  student  by  the  instructor  will  insure  the 
rapid  acquirement  of  skill  in  the  use  of  the 
'°  instrument.     In  what  follows,  the  four  scales  of 

o.  the  slide  rule  are  designated  fromi  top  to  bottom 

of  the  rule.  A,  B,  C,  D,  respectively.     The  ends 
of  the  scales  are  called  the  indices. 
^       ^        An  ordinary   10-inch  slide  rule  should  give 
M     Pi    results    accurate    to    three  significant  figures, 
•«*'     -^     which  is  accurate  enough  for  most  of  the  pur- 

^    poses  of  applied  science. 
^,       -^        An  exaggerated  idea  sometimes  prevails  con- 
's    cerning  the  degree  of  accuracy  required  by  work 
b    in  science  or  in  applied  science.     Many  of  the 
-^       ^    fundamental  constants  of  science,  upon  which  a 
^     large  number  of  other  results  depend,  are  known 
^     only  to  three  decimal  places.     In  such  cases 
I.     greater  than  three  figure  accuracy  is  impossible 
^       o     even  if  desired.     In  other  cases  greater  accuracy 
^    is  of  no  value  even  if  possible.     The  real  desid- 
Ph     eratum  in  computed  results  is,  first,  to  know  by  a 
suitable  check  that  the  work  of  computation  is  correct, 
and,  second,  to  know  to  what  order  or  degree  of 
?  accuracy  both  the  data  and  the  result  are  dependable. 

2  The  absurdity  of  an  undue  number  of  decimal 

i  places  in  computation  is  illustrated  by  the  orig- 

^.  inal  tables  of  logarithms,  which  if  now  used 

'^  would  enable  one  to  compute  from  the  radius 

"  of  the  earth,  the  circumference  correct  to  1  /10,000 

"  part  of  an  inch. 

;3  '^  The  following  matters  should  be  emphasized 

in  the  use  of  the  sHde  rule: 


05 


§149]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   249 

(1)  All  numbers  for  the  purpose  of  computation  should  be  con- 
sidered as  given  with  the  first  figure  in  units  place.  Thus  517 
X  1910  X  0.024  should  be  considered  as  5.17  X  1.19  X  2.4  X 
102  X  10^  X  10-2.  The  result  should  then  be  mentally  approxi- 
mated (say  24,000)  for  the  purpose  of  locating  the  decimal  point, 
and  for  checking  the  work. 

(2)  A  proportion  should  always  be  solved  by  one  setting  of  the 
slide. 

(3)  A  combined  product  and  quotient  like 

aXbXcX d 
rXs  xT 
should  always  be  solved  as  follows: 

Place  runner  on  a  of  scale  D. 

Set  r  of  scale  C  to  a  of  scale  D; 

Runner  to  6  of  C; 

s  of  (7  to  runner; 

Runner  to  c  of  C; 

<  of  C  to  runner; 

at  (i  of  C  find  on  D  the  significant  figures  of  the  result. 

(4)  The  runner  must  be  set  on  the  first  half  of  A  for  square 
roots  of  odd  numbered  numbers,  and  on  the  second  half  of  A  for 
the  square  roots  of  eveii  numbered  numbers. 

(5)  Use  judgment  so  as  to  compute  results  in  most  accurate 
manner — thus  instead  of  computing  264/233,  compute  31  /233  and 
hence  find  264/233  =  1  +  31/233.1 

(6)  Besides  checking  by  mental  calculation  as  suggested  in  (1) 
above,  also  check  by  computing  several  neighboring  values  and 
graphing  the  results  if  necessary.  Thus  check  5.17  X  1.91  X  2.4 
by  computing  both  5.20  X  19.2  X  2.42  and  5.10  X  1.90  X  2.38. 

Exercises 
Compute  the  following  on  the  slide  rule. 

1.  3.12  X  2.24;  1.89  X  4.25;  2.88  X  3.16;  3.1  X  236. 

2.  8.72/2.36;  4.58/2.36;  6.23/2.12;  10/3.14. 

3.  32.5  X  72.5;  0.000116  X  0.00135;  0.0392/0.00114. 

4.  3,967,000  --  367,800,000. 
6.54  X  42.6.    8.75  X  5.25 

^*         32.5        '  32.3 

1  Show  by  trial  that  this  gives  a  more  accurate  result. 


250        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§149 
^    78  5  X  36.6  X  20.8 


5.75  X  29.5 
6.46  X  57.5  X  8.55 


3.26  X  296X0.642 
8.  Solve  the  proportion 

x:1.72:  =  :4.14:  V2^ 
where  g  =  32.2  and  h  =  78.2. 
^    ^  ^     VlTi  X  1.41. 

10.  The  following  is  an  approximate  formula  for  the  area  of  a  seg- 
ment of  a  circle: 

A  =  hy2c  +  2c/i/3 
where  c  is  the  length  of  the  chord  and  h  is  the  altitude  of  the  segment. 
Test  this  formula  for  segments  of  a  circle  of  unit  radius,  whose  arcs 
are  7r/3,  7r/2,  and  tt  radians,  respectively. 

11.  Two  steamers  start  at  the  same  time  from  the  same  port;  the 
first  sails  at  12  miles  an  hour  due  south,  and  the  second  sails  at  16 
miles  an  hour  due  east.  Find  the  bearing  of  the  first  steamer  as  seen 
from  the  second  (1)  after  one  hour,  (2^  after  two  hours,  and  compute 
their  distances  apart  at  each  time. 

The  following  exercises  require  the  use  of  the  data  printed  herewith. 
An  ''acre-foot "  means  the  quantity  of  water  that  would  cover  1 
acre  1  foot  deep,  "Second-foot"  means  a  discharge  at  the  rate  of  1 
cubic  foot  of  water  per  second.  By  the  "run-off"  of  any  drainage  area 
is  meant  the  quantity  of  water  flowing  therefrom  in  its  surface  stream 
or  river,  during  a  year  or  other  interval  of  time. 

1  square  mile  =  640  acres 

1  acre  =^  43,560  square  feet. 

1  day  =  86,400  seconds. 

1  second  foot  =  2  acre  feet  per  day. 

1  cubic  foot  =  7|  gallons. 

1  cubic  foot  water  =  62|  pounds  water. 

1  h.p.  =  550  foot  pounds  per  second. 

450  gallons  per  minute  =  1  second  foot. 

Each  of  the  following  problems  should  be  handled  on  the  slide  rule  as 
a  continuous  piece  of  computation. 

12.  A  drainage  area  of  710  square  miles  has  an  annual  run-off  of 
120,000  acre  feet.  The  average  annual  rainfall  is  27  inches.  Find 
what  percent  of  the  rainfall  appears  as  run-off. 

13.  A  centrifugal  pump  discharges  750  gallons  per  minute  against 


§150]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   251 

a  total  lift  of  28  feet.  Find  the  theoretical  horse  power  required. 
Also  daily  discharge  in  acre  feet  if  the  pump  operates  fourteen  hours 
per  day. 

14.  What  is  the  theoretical  horse  power  represented  by  a  stream 
discharging  550  second  feet  if  there  be  a  fall  of  42  feet? 

15.  A  district  containing  25,000  acres  of  irrigable  land  is  to  be  sup- 
plied with  water  by  means  of  a  canal.  The  average  annual  quantity 
of  water  required  is  3^  feet  on  each  acre.  Find  the  capacity  of  the 
canal  in  second  feet,  if  the  quantity  of  water  required  is  to  be  delivered 
uniformly  during  an  irrigation  season  of  five  months. 

16.  A  municipal  supply  amounts  to  35,000,000  gallons  per  twenty- 
four  hours.     Find  the  equivalent  in  cubic  feet  per  second. 

17.  A  single  rainfall  of  3.9  inches  on  a  catchment  area  of  210  square 
miles  is  found  to  contribute  17,500  acre  feet  of  water  to  a  storage 
reservoir.     The  run- off  is  what  percent  of  the  rainfall  in  this  case? 


.1  .2  .3  .4  .5  .6  .7         .8  .9         1.0"" 

Fig.   101. — The  Theory  of  the  Use  of  Semi-logarithmic  Paper. 


150.  Semi-logarithmic  Coordinate  Paper.  Fig.  101  represents 
a  sheet  of  rectangular  coordinate  paper,  on  which  ON  has  been 
chosen  as  the  unit  of  measure.     Along  the  right-hand  edge  of  this 


252        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§150 

sheet  is  constructed  a  logarithmic  scale  LM  of  the  type  discussed 
in  §  148,  i.e.,  any  number,  say  4,  on  the  scale  LM  stands  opposite 
the  logarithm  of  that  number  (in  the  case  named  opposite  0.6021) 
on  the  uniform  scale  ON. 

Let  us  agree  always  to  designate  by  capital  letters  distances 
measured  on  the  uniform  scales,  and  by  lower  case  letters  dis- 
tances measured  on  the  logarithmic  scale.  Thus  Y  will  mean  the 
ordinate  of  a  point  as  read  on  the  scale  ON,  while  y  will  mean  the 
ordinate  of  a  point  as  read  on  the  scale  LM.  In  other  words,  we 
agree  to  plot  a  function,  using  logarithms  of  the  values  of  the 
function  as  ordinates  and  the  natural  values  of  the  argument  or 
variable  as  abscissas. 

Let  PQ  be  any  straight  line  on  this  paper,  and  let  it  be  required 
to  find  its  equation,  referred  to  the  uniform  x-scale  OL  and  the 
logarithmic  i/-scale  LM.     We  proceed  as  follows: 

The  equation  of  this  line,  referred  to  the  uniform  X-axis  OL 
and'  the  uniform  7- axis  ON,  where  0  is  the  origin,  is 

Y  =  mX  -\-B 

m  being  the  slope  of  the  line,  and  B  its  ^/-intercept.  Now,  for  the 
line  PQ,  m  =  0.742  and  B  =  0.36,  so  that  the  equation  of  PQ  is 

Y  =  0.742X  -f  0.36  (1) 

To  find  the  equation  of  this  curve  referred  to  the  scales  LM  and 
OL,  it  is  only  necessary  to  notice  that 

Y  =  logy 
so  that  we  obtain: 

log  y  =  0.742X  -h  0.36  (2) 

The  intercept  0.36  was  read  on  the  scale  ON,  and  is  therefore  the 
logarithm  of  the  number  corresponding  to  it  on  the  scale  LM. 
That  is,  0.36  =  log  2.30.  Substituting  this  value  in  equation  (2) 
we  obtain: 

log  y  =  0.742.T  +  log  2.30 

which  may  be  written 

log  y  -  log  2.30  =  0.742a; 
or, 

log  2I0  =  0.742. 


§150]  LOGARITHMIC  AND  EXPONENTAIL  FUNCTIONS   253 
On  changing  to  exponential  nota^tion  this  becomes: 


or, 


o»  00 


D'*M 


y 


100-74'2x 


2.30 

y  =  2.30(100  '42x) 


(3) 


CD  US 


m  n 


N°C 


TTTTT  ITIIII 


:z: 


z 


^ 


A      L     0.1  0.2  0.3  0.4  0.5  O.d  0.7  0.8  0.9        l.OB 

Semi  Logarrthmlc  Paper 

Fig.    102. — Illustration  of  Squared  Paper,  Form  Mb.     The  finer  rulings  of 
Form  Mb  have  been  omitted  in  Fig.  102. 


In  general,  if  the  equation  of  a  straight  line  referred  to  the 
scales  OL  and  ON  is 

Y  =  mX  +  B  (4) 

its  equation  referred  to  the  scales  OL  and  LM  may  be  obtained  by 
replacing  Y  by  log  y  and  B  by  log  h  in  the  manner  described  above, 
giving 

log  y  =  mx  +  log  h  (5) 


254        ELEMENTARY  MATHEMATICAL  ANALYSIS      l§150 

which,  as  above,  may  be  reduced  to  the  form 

y  =  blO-^  (6) 

This  is  the  general  equation  of  the  exponential  curve.     Hence: 

Any  exponential  curve  can  be  represented  by  a  straight  line,  provided 

ordinates  are  read  from  a  suitable  logarithmic  scale,  and  abscissas 

are  read  from  a  uniform  scale. 


Jl  0.1         0.2         0.3  0.4  0.5  0.6  0.7  0.8         0.9         1.0  B 

Semi  Logarithmic  Paper 

Fig.    103. — Exponential   Curves   on   Form   M5.     The   curve  —  ...    is 
y  =  10-3*;  _      is  2/  =  10-2*;  —   .    .    is  2/  =  103*. 

Fig.  102  represents  the  same  line  PQ  {y  =  (2.30)10o  742x)^  as 
Fig.  101.  The  two  figures  differ  only  in  one  respect:  in  Fig.  101 
the  rulings  of  the  uniform  scale  ON  are  extended  across  the  page, 
while  in  Fig.  102  these  rulings  are  replaced  by  those  of  the  scale 
LM. 

Coordinate  paper  such  as  that  represented  by  Fig.  102  is  known 


§150]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   255 

as  semi-logarithmic .  paper.     It  affords  a  convenient  coordinate 
system  for  work  with  the  exponential  function. 

Every  point  on  PQ  (Fig.  102)  satisfies  the  exponential  equation 
y  =  2.30(10°  742.) 

Thus,  in  the  case  of  the  point  R, 

3.98   =    2.30(100-742)0.320 

=  2.30(100-238) 

The  slope  of  any  line  on  the  semi-logarithmic  paper  may  be  read 
or  determined  by  means  of  the  uniform  scales  BC  and  AB  of  form 
Mb.  The  scale  AD  of  form  Af5  is  the  scale  of  the  natural  loga- 
rithms, so  that  any  equation  of  the  form  y  =  e"*^  can  be  graphed 
at  once  by  the  use  of  this  scale.  Thus,  the  line  y  =  e""  (Fig. 
103)  passes  through  the  point  A  or  (0,  1),  and  a  point  on  BC  op_ 
posite  the  point  marked  1.0  on  AD.  Note  that  1.0  on  scale  AD 
2.718  on  the  non-uniform  scale  of  the  main  body  of  the  paper 
and  0.4343  on  the  scale  BC  all  fall  together,  as  they  should. 

To  draw  the  line  y  =  10~^,  the  corner  D  of  the  plate  may  be 
taken  as  the  point  (0,  1).  On  the  line  drawn  once  across  the  sheet 
representing  y  =  lO"*"",  y  has  a  range  between  1  and  10  only. 
To  represent  the  range  of  y  between  10  and  100,  two  or  more  sheets 
of  form  M5  may  be  pasted  together,  or,  preferably,  the  continua- 
tion of  the  line  may  be  shown  on  the  same  sheet  by  suitably 
changing  the  numbers  attached  to  the  scales  AB  and  BC.  Thus 
Fig.  103  shows  in  this  manner  y  =  10^^  and  y  =  10^^. 

Remember  that  the  line 

y  =  blO--  (7) 

passes  through  the  point  (0,  h)  with  slope  m.     Note  that  , 

I,    ^    ^Qm{x  -  a)  (g) 

D 

passes  through  the  point  {a,  b)  with  slope  m. 

Exercises 

On  semi-logarithmic  paper  draw  the  following: 

1.  y  =  103*,  y  =  102*,  y  =  10*,  y  =  10-*,  y  =  10  "2*,  y  =  10-»*. 

2.  y  =  e^*,  y  =  e',  y  =  e~',  y  =  e"**. 

3.  Sx  =  log  y,  (l/2)x  =  log  y. 


256        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§151 

4.  ?/  =  10^/^  y  =  10^/^0. 

5.  Graph?/  =  2(10)* and  |  =  10* "3. 

151.  The  Compound  Interest  Law.     Logarithmic  Increment. 

The  law  expressed  by  the  exponential  curve  was  called  by  Lord 
Kelvin  the  compound  interest  law  and  since  that  time  this  name 
has  been  generally  used.  It  is  recalled  that  the  exponential  curve 
was  drawn  by  using  ordinates  equal  to  the  successive  terms  of 
a  geometrical  progression  which  are  uniformly  spaced  along  the 
X-axis;  since  the  amount  of  any  sum  at  compound  interest  is  given 
by  a  term  of  a  geometrical  progression,  it  is  obvious  that  a  sum  at 
compound  interest  accumulates  by  the  same  law  of  growth  as  is 
indicated  by  a  set  of  uniformly  spaced  ordinates  of  an  expo- 
nential curve;  hence  the  term  "compound  interest  law,"  from 
this  superficial  view,  is  appropriate.  The  detailed  discussion 
that  follows  will  make  this  clear: 

Let  $1  be  loaned  at  r  percent  per  annum  compound  interest. 
At  the  end  of  one  year  the  amount  is:  (1  -f  r/100). 
At  the  end  of  two  years  the  amount  is:  (1  +  r /lOO)^ 
and  at  the  end  of  t  years  it  is:  (1  +  r/100)'. 
If  interest  be  compounded  semi-annually,  instead  of  annually, 
the  amount  at  the  end  of  t  years  is:  (1  +  r/200)^' 
and  if  compounded  monthly  the  amount  at  the  end  of  the  same 
period  is:  (1  +  r/1200)i2< 

or  if  compounded  n  times  per  year  y  =  {I  -\-  r/lOOn)"' 
where  t  is  expressed  in  years.  Now  if  we  find  the  limit  of  this 
expression  as  n  is  increased  indefinitely,  we  will  find  the  amount  of 
principle  and  interest  on  the  hypothesis  that  the  interest  was 
compounded  continuously.  For  convenience  let  r/100n=l/it. 
Then: 

7/  =  (1-f  lluyrtn^^.  (1) 

where  the  limit  is  to  be  taken  as  w  or  n  becomes  infinite.     Calling 

(l  +  l^)"=/(^)  (2) 

and  expanding  by  the  binomial  theorem  for  any  integral  value  of 

Uj  we  obtain: 

/(«)  =  !  +  «(!/«)  +  "^"2^^      i+    •    •    ■ 

=  1  +  1  +  (1  -  1/m)/2!  +  (1  -  1/«)(1  -  2/«)/3!  +  .    .    .   (3) 


§151]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   257 

In  the  calculus  it  is  shown  that  the  limit  of  this  series  as  u  becomes 
infinite  is  the  limit  of  the  series 

1  +  1 +  1/2!  + 1/3!+  ...  (4) 

The  limit  of  this  series  is  easily  found;  it  is,  in  fact,  the  Napierian 
base  e.  It  is  shown  in  the  calculus  that  the  restriction  that  u 
shall  be  an  integer  may  be  removed,  so  that  the  limit  of  (3)  may 
be  found  when  w  is  a  continuous  variable. 

It  is  easy  to  see  that  the  limit  of  (4)  is  >  2^  and  <  3.  The  sum 
of  the  first  three  terms  of  the  series  (4)  equals  2\\  the  rest  of  the 
terms  are  positive,  therefore  e  >  2|.  The  terms  of  the  series  (4), 
after  the  first  three,  are  also  observed  to  be  less,  term  for  term,  than 
the  terms  of  the  progression: 

(1/2)2 +  (1/2)3+  _    ,  (5) 

But  this  is  a  geometrical  progression  the  limit  of  whose  sum  is  1/2. 
Therefore  (3)  is  always  less  than  2|  +  |  or  3.  The  value  of  e  is 
readily  approximated  by  the  following  computation  of  the  first 
8  terms  of  (4) : 

2.00000  =1  +  1 

3 10. 50000  =  1/2! 

4|0. 16667  =  1/3! 

5|0. 04167  =  1/4! 

6|0. 00833  =  1/5! 

7 10. 00139  =  1/6! 
0.00020  =  1/7! 
Sum  of  8  terms  =  2. 71826 

The  value  of  e  here  found  is  correct  to  four  decimal  places. 

Returning  to  equation  (1)  above,  the  amount  of  $1  at  r 
percent  compound  interest  compounded  continuously  is: 

y   =   gri/100 

Thus  $100  at  6  percent  compound  interest,  compounded  annually, 
amounts,  at  the  end  of  ten  years,  to 

y  =  100(1.06)10  =  $179.10 
The  amount  of  $100  compounded  continuously  for  ten  years  is 

y  =  lOOe" «  =  $182.20 
The  difference  is  thus  $3.10. 

17 


258        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§151 

The  compound  interest  law  is  one  of  the  important  laws  of 
nature.  As  previously  noted,  the  slope  or  rate  of  increase  of  the 
exponential  function 

y  =  ae^* 

at  any  point  is  always  proportional  to  the  ordinate  or  to  the  value 
of  the  function  at  that  point.  Thus  when  in  nature  we  find  any 
junction  or  magnitude  that  increases  at  a  rate  proportional  to  itself 
we  have  a  case  of  the  expontential  or  compound  interest  law. 
The  law  is  also  frequently  expressed  by  saying,  as  has  been  re- 
peatedly stated  in  this  book,  that  the  first  of  two  magnitudes  varies 
in  geometrical  progression  while  a  second  magnitude  varies  in  arith- 
metical progression.  A  familiar  example  of  this  is  the  increased 
friction  as  a  rope  is  coiled  around  a  post.  A  few  turns  of  the  haw- 
sers about  the  bitts  at  the  wharf  is  sufficient  to  hold  a  large  ship, 
because  as  the  number  of  turns  increases  in  arithmetical  progres- 
sion, the  friction  increases  in  geometrical  progression.  Thus  the 
following  table  gives  the  results  of  experiments  to  determine  what 
weight  could  be  held  up  by  a  one-pound  weight,  when  a  cord 
attached  to  the  first  weight  passed  over  a  round  peg  the  number 
of  times  shown  in  the  first  column  of  the  table: 


n  =  number  of 

turns  of  the  cord 

on  the  peg 

w  =  weight  just  held 
in  equilibrium  by 
one-pound  weight 

Logs  of  preceding 
numbers 

d  =  logarithmic 
[increment 

1/2 
1 

u 

2 
3 

1.6 
3.0 
5.1 
8.0 
14.0 
23.0 

0.204 
0.477 
0.709 
0.903 
1.146 
1.362 

0.273 
0.231 
0.195 
0.243 
0.216 

Average  logarithmic  increment  = 


0.23 


If  the  weights  sustained  were  exactly  in  geometrical  progression, 
their  logarithms  would  be  in  arithmetical  progression.  The  test 
for  this  fact  is  to  note  whether  the  differences  between  logarithms 
of  successive  values  are  constant.  These  differences  are  known 
as  logarithmic  increments  or  in  case  they  are  negative,  as  loga- 
rithmic decrements.  In  the  table  the  logarithmic  increments 
fluctuate  about  the  mean  value  0.23. 


§152]   LOGARITHMIC  AND  EXPONENTAIL  FUNCTIONS  259 

The  equation  connecting  n  and  w  is  of  the  form 
w  =  10"  ^"*  or  n  =  m  log  w 
By  graphing  columns  1  and  3  on  squared  paper,  the  value  of  m  is 
determined  and  we  find 

w  =  10o-45«  or  71  =  2.2  log  2 
Another  way  is  to  graph  columns  1  and  2  on  semi-logarithmic 
paper. 

An  interesting  example  of  the  compound  interest  law  is  Weber's 
law  in  psychology,  which  states  that  if  stimuU  are  in  geometrical 
progression,  the  sense  perceptions  are  in  arithmetical  progression. 

152.  Modulus  of  Decay,  Logarithmic  Decrement.  In  a  very 
large  number  of  cases  in  nature  the  "compound  interest"  law 
appears  as  a  decreasing  function  rather  than  as  an  increasing 
function,  so  that  the  equation  is  of  the  form 

y  =  ae-*^  (1) 

where  —  b  is  essentially  negative.  The  following  are  examples  of 
this  law: 

(1)  If  the  thickness  of  panes  of  glass  increase  in  arithmetical 
progression,  the  amount  of  light  transmitted  decreases  in  geo- 
metrical progression.     That  is,  the  relation  is  of  the  form 

L  =  ae-f''  (2) 

where  t  is  the  thickness  of  the  glass  or  other  absorbing  material 
and  L  is  the  intensity  of  the  light  transmitted.  Since  when  t  =  0 
the  light  transmitted  must  have  its  initial  intensity,  Lq,  (2) 
becomes 

L  =  Loe-^'  (3) 

The  constant  b  must  be  determined  from  the  data  of  the  problem. 
Thus,  if  a  pane  of  glass  absorbs  2  percent  of  the  incident  light, 

Lo  =  100,  L  =  98  for  ^  =  1, 
then:  98  =  lOOe"'' 

or  log  98  -  log  100  =  -  6  log  e. 

Therefore:  b  =  J'^^f  =  0.02 

The  light  transmitted  by  ten  panes  of  glass  is  then 
Lio  =  100e-io(oo2)  =  lOOe-o  2 


260        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§152 

or,  by  the  table  of  §146, 

Lio  =  100/1.2214  =  82  percent 

(2)  Variation  in  atmospheric  pressure  with  the  altitude  is 
usually  expressed  by  Halley's  Law: 

p  =  760e-''/8ooo 

where  h  is  the  altitude  in  meters  above  sea  level  and  p  is  the  at- 
mospheric pressure  in  millimeters  of  mercury.  See  §147,  Exer- 
cises 5,  6,  7. 

(3)  Newton's  law  of  cooling  states  that  a  body  surrounded  by  a 
medium  of  constant  temperature  loses  heat  at  a  rate  proportional 
to  the  difference  in  temperature  between  it  and  the  surrounding 
medium.  This,  then,  is  a  case  of  the  compound  interest  law. 
If  0  denotes  temperature  of  the  cooling  body  above  that  of  the 
surrounding  medium  at  any  time  t,  we  must  have 

6  =  ae-«" 
The  constant  a  must  be  the  value  of  6  when  f  =  0,  or  the  initial 
temperature  of  the  body. 

Exercises 

1.  A  thermometer  bulb  initially  at  temperature  19°. 3  C.  is  exposed 
to  the  air  and  its  temperature  d  observed  to  be  14°.2  C.  at  the  end  of 
twenty  seconds.  If  the  law  of  cooling  be  given  by  0  =  doe'^^  where 
t  is  the  time  in  seconds,  find  the  value  of  6  and  b. 

Solution:  The  condition  of  the  problem  gives  6  =  19.3  when 
t  =  0,  hence  do  =  19.3.     Also,  14.2  =  19.36-2*.     This  gives 

log  19.3  -  20&log  e  =  log  14.2 
from  which  6  can  be  readily  computed. 

2.  If  1|  percent  of  the  incident  light  is  lost  when  light  is  directed 
through  a  plate  of  glass  0.3  cm.  thick,  how  much  light  would  be 
lost  in  penetrating  a  plate  of  glass  2  cm.  thick? 

3.  Forty  percent  of  the  incident  light  is  lost  when  passed  through 
a  plate  of  glass  2  inches  thick.  Find  the  value  of  a  in  the  equation 
L  =  LqB  -'*'  where  t  is  thickness  of  the  plate  in  inches,  L  is  the  per- 
cent of  light  transmitted,  and  Lq  =  100. 

4.  As  I  descend  a  mountain  the  pressure  of  the  air  increases  each 
foot  by  the  amount  due  to  the  weight  of  the  layer  of  air  1  foot  thick. 
As  the  density  of  this  layer  is  itself  proportional  to  the  pressure,  show 
that  the  pressure  as  I  descend  must  increase  by  the  compound  interest 
law. 


§153]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   261 

5.  Power  is  transmitted  in  a  clock  through  a  train  of  gear  wheels 
n  in  number.  If  the  loss  of  power  in  each  pair  of  gears  is  10  percent, 
draw  a  curve  showing  the  loss  of  power  at  the  nth  gear. 

Note:  The  graphical  method  of  §124,  Figs.  88,  89,  may  appro- 
priately be  used. 

6.  Given  that  the  intensity  of  light  is  diminished  2  percent  by 
passing  through  one  pane  of  glass,  find  the  intensity  /  of  the  light 
after  passing  through  n  panes. 

7.  The  number  of  bacteria  per  cubic  centimeter  of  culture  increases 
under  proper  conditions  at  a  rate  proportional  to  the  number  present . 
Find  an  expression  for  the  number  present  at  the  end  of  time  t  if  there 
are  1000  per  cubic  centimeter  present  at  time  zero,  and  8000  per 
cubic  centimeter  present  at  time  10. 

8.  The  temperature  of  a  body  cooling  according  to  Newton's  law 
fell  from  30°  to  18°  in  six  minutes.  Find  the  equation  connecting  the 
temperature  of  the  body  and  the  time  of  cooling. 

153.  Empirical  Curves  on  Semi-logarithmic  Coordinate  Paper. 
One  of  the  most  important  uses  of  semi-logarithmic  paper  is  in 
determining  the  functional  relation  between  observed  data,  when 
such  data  are  connected  by  a  relation  of  the  exponential  form. 
Suppose,  for  example,  that  the  following  are  the  results  of  an 
experiment  to  determine  the  law  connecting  two  variables  x  and  y : 


X 

0.04 

0.18 

0.36 

0.51 

0.685 

0.833 

0.97 

y 

5.3 

4.4 

3.75 

3.1 

2.6 

2.33 

1.9 

If  the  equation  connecting  x  and  y  is  of  the  exponential  form,  the 
points  whose  coordinates  are  given  by  corresponding  values  of  x 
and  y  in  the  table  will  lie  in  a  straight  line,  except  for  such  slight 
errors  as  may  be  due  to  inaccuracies  in  the  observations.  Plotting 
the  points  on  semi-logarithmic  coordinate  paper,  we  find  that  they 
lie  nearly  on  the  line  PQ  (Fig.  104).  Assuming  that,  if  the  data 
were  exact,  the  points  would  lie  exactly  on  this  line,^  we  may  pro- 

1  We  would  not  be  at  liberty  to  make  such  an  assumption  if  the  variation  of  the 
points  away  from  the  line  was  of  a  character  similar  to  that  represented  by 
the  dots  near  the  top  of  Fig.  104.  These  points,  although  not  departing  greatly 
from  the  line  shown,  depart  from  it  systematically.  That  is,  they  lie  below  it  at 
each  end,  and  above  it  in  the  center,  seeming  to  approximate  a  curve,  (such  as  the 
one  shown  dotted)  more  nearly  than  the  line.  The  points  arranged  about  the  line 
PQ  depart  as  far  from  that  line  as  do  the  points  above  the  higher  line,  but  they 
do  not  depart  systematically,  as  if  tending  to  lie  along  a  smooth  curve.  When  points 
arrange  themselves  as  at  the  top  of  Fig.  104,  one  must  infer  that  the  relation  con- 
necting the  given  data  is  not  exponential  in  character. 


262        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§153 

ceed  to  determine  the  equation  of  this  line  as  approximately  repre- 
senting the  relation  between  x  and  y. 

It  is  easy  to  find  the  equation  of  such  a  line  referred  to  the  uni- 
form scales  AB  and  BC  of  form  ilf  5.  We  may  imagine  that  all 
rulings  are  erased  and  replaced  by  extensions  of  the  uniform  AB 
scale,  as  in  Fig.  101.     The  equation  of  the  line  PQ  is  then 


y  =  mX  +  5 


(1) 


D-^'M   => 


J^'" 

" 

""""' 

IIMIIIII 

'""MM 

iiimlOl 

jg 

^-^ 

9: 

3 

*K 

; 

Is 

V^ 

8  r 

i 

^ 

17 

^^ 

^ 

7  = 

^N 

1  6 

^ 

■V 

C  ~ 

i  p 

N 

^5^ 

1 

\ 

34 

^ 

^-v^ 

4E 

-1 

1 

1  3 

jv^^ 

8  - 

' 

^•^^ 

1  2 

^ 

^    o- 

1 
J 

1  1 

1 1 

1 1 1 1  r  1  1 1 1 

A     L       01  0-2  0.3  0.4  0.5  0.6  0.7  0.8  0.9        l.QB 

Semi  Logarithmic  Paper 

Fig.  104. — Empirical  Equations  Determined  by  Use  of  Form  Mb. 


where  m  is  the  slope,  and  B  is  the  ^/-intercept.  Now,  for  PQ, 
m=  -  0.447  and  B  =  0.730  =  log  5.37.  Equation  (1)  of  PQ 
becomes,  therefore: 

7  =  -  0.447X  +  0.730 


§154]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   263 

or,  replacing  Y  by  log  y  and  0.730  by  log  5.37,  in  order  to  refer 
the  curve  to  the  scales  AB  and  LM, 

log  y  -  log  5.39  =  -  0.447x 
whence 

y  =  5.39(10-o-447x)  (2) 

If  it  is  desired  to  express  the  relation  to  the  base  e  instead  of 
base  10,  we  may  note  10  =  e2.3026  (§143,  equation  (1)),  or,  sub- 
stituting in  (2), 

y  =  5.39  (e2  303)-o.i47x 

=  5.39  e-i-029-  (3) 

The  same  result  might  have  been  obtained  directly  by  use  of 
the  uniform  scale  AD,  at  the  left  of  form  M5.  This  scale  is  so 
constructed  that  the  length  I  on  AB  corresponds  to  the  length 
2.3026  on  AD.  Now,  we  know  that  e2.3026  =  iq,  hence  we  may 
replace  10  in  10""^  by  e  if  we  make  m  in  lO'"^'  2.3026  times  as 
great  as  before.  This  is  readily  done  by  measuring  the  slope  of 
PQ  by  the  use  of  the  uniform  scale  AD  instead  of  the  uniform 
scale  BC.  Computing  the  slope  of  PQ  by  use  of  the  scale  AD 
we  find: 

F  of  Q  =  0.653 

FofP  =  1.681 

Difference  =  -  1.028 
Since  AB  =  1,  this  is  the  slope  of  the  line,  measured  to  the  scale 
AD,  and  is  therefore  the  value  of  m  in  the  equation 

y  =  ae'"^  (4) 

Hence  the  equation  of  PQ  is 

y  =  5.39e-io28x 

which  agrees  with  the  equation  previously  obtained. 

154.  Change  of  Scale  on  Semi-logarithmic  Paper.  A  sheet  of 
semi-logarithmic  paper,  form  M5,  is  a  square.  If  sheets  of  this 
paper  be  arranged  "checker-board  fashion"  over  the  plane,  then 
the  vertical  non-uniform  scale  will  be  a  repetition  of  the  scale  LM, 
Fig.  104,  except  that  the  successive  segments  of  length  LM  must 
be  numbered  1,  2,  3,  .  .  . ,  9  for  the  original  LM,  then  10,  20, 
30,  .  .  . ,  90  for  the  next  vertical  segment  of  the  checker-board, 
then  100,  200,  300,  .  .  . ,  900,  for  the  next,  etc.  It  is  obvious, 
therefore,  that  the  initial  point  A  of  a  sheet  of  semi-logarithmic 


264        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§154 

paper  may  be  said  to  have  the  ordinate  1,  or  10,  or  100,  etc.,  or 
10~\  10~2j  etc.,  as  may  be  most  convenient  for  the  particular 
graph  under  consideration.  The  horizontal  scale  being  a  uniform 
scale,  any  values  of  x  may  be  plotted  to  any  convenient  scale  on 
it,  as  when  using  ordinary  squared  paper.  However,  if  the  hori- 
zontal unit  of  length  (the  length  AB,  form  Mb)  be  taken  as  any 
value  different  from  unity,  then  the  slope  m  of  the  line  PQ  drawn 
on  the  semi-logarithmic  paper  can  only  be  found  by  dividing  its 
apparent  slope  by  the  scale  value  of  the  side  AB.  That  is,  the 
correct  value  of  m  in 

y  =  alO'"^ 
is,  in  all  cases, 

_  apparent  slope  of  PQ 

~      scale  value  oi  AB 

The  "apparent  slope"  of  PQ  is  to  be  measured  by  applying  any 
convenient  uniform  scale  of  inches,  centimeters,  etc.,  to  the 
horizontal  and  vertical  sides  of  a  right  triangle  of  which  PQ  is  the 
hypotenuse. 

Exercises 

1.  A  thermometer  bulb  initially  at  temperature  19.3°  C.  is  exposed 
to  the  air  and  its  temperature  d  noted  at  various  times  t  (in  seconds) 
as  follows : 


0         20         40         60       80     100     120 


19.3      14.2      10.4      7.6     5.6      4.1     3.0 
Plot  these  results  on  semi-logarithmic  paper  and  test  whether  or  not 
e  follows  the  compound  interest  law.     If  so,  determine  the  value  of 
^0  and  h  in  the  equation  d  =  d^e  "*^     Note  that  the  last  point  given 
by  the  table,  namely  t  =  120,  0  =  3.0,  goes  into  a  new  square  if  the 
scale  AB  be  called  0—100.     If  the  scale  AB  be  called  0—200  then  all 
entries  can  appear  on  a  single  sheet  of  form  M5. 
2.  Graph  the  following  on  semi-logarithmic  paper: 
1/2       1        1|        2         2|  3 


1.6     3.0     5.1     8.0      14.0     23.0 

Show  that  the  equation  connecting  n  and  w  is  w  —  10° •^^'*. 

Suggestion:    The  scale  AB,  form  Af5,  may  be  called  0 — 5  for  the 
purpose  of  graphing  n. 


§155]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   265 


5.  Graph  the  following  on  semi-logarithmic  paper,  and  find  the 
equation  connecting  n  and  w. 


0.2 

0.4      0.6 

0.8 

1.0 

1.2 

1.4 

1.6 

2.60 

3.41     4.45 

5.75 

7.56 

9.85 

1.30 

16.6 

4.  A  circular  disk  is  suspended  by  a  fine  wire  at  its  center.  When 
at  rest  the  upper  end  of  the  wire  is  turned  by  means  of  a  supporting 
knob  through  30°.  The  successive  angles  of  the  torsional  swings  of 
the  disk  from  the  neutral  point  are  then  read  at  the  end  of  each  swing 
as  follows : 


Swing  number 

1 

2 

3 

4 

5 

6 

7 

Angle 

26°.4 

23°.2 

20°.5 

18°.0 

15°.9 

14°.0 

12°.3 

Show  that  the  angle  of  the  successive  swings  follows  the  compound 
interest  law  and  find  in  at  least  two  different  ways  the  equation 
connecting  the  number  of  the  swing  and  the  angle.  Show  by  the 
slide  rule  that  the  compound  interest  law  holds. 

155.  The  Power  Function  Compared  with  the  Exponential 
Function.  It  has  been  emphasized  in  this  book  that  the  fuD da- 
mental  laws  of  natural  science  are  three  in  number,  namely:  (1) 
the  parabolic  law,  expressed  by  the  power  function  y  =  ax" 
where  n  may  be  either  positive  or  negative;  (2)  the  harmonic  or 
periodic  law,  y  =  asm  nx,  which  is  fundamental  to  all  periodically 
occurring  phenomena;  and  (3)  the  compound  interest  law  dis- 
cussed in  this  chapter.  While  there  are  other  important  laws  and 
functions  in  mathematics,  they  are  secondary  to  those  expressed 
by  these  fundamental  functions.  The  second  of  the  functions 
above  named  will  be  more  fully  discussed  in  the  chapter  on  waves. 
The  discussion  of  the  compound  interest  law  should  not  be  closed 
without  a  careful  comparison  of  power  functions  and  exponential 
functions. 

The  characteristic  property  of  the  power  function 


y  =  ax" 


(1) 


is  that  as  x  changes  by  a  constant  factor,  y  changes  by  a  constant 
factor  also.    Take 

y  =  ax-  =  fix)  (2) 


266        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§155 

Let  X  change  by  a  constant  factor  m,  so  that  the  new  value  of  x 
is  mx.     Call  1/  the  new  value  of  y.     Then 

y'  =  a{mx) "  =  /(mx)  (3) 

That  is: 

y        a{mxY 

,  =  -^ — ^-  =  m"  (4) 

y'  ax"  ^  ' 

which  shows  that  the  ratio  of  the  two  y's  is  independent  of  the  value 
of  x  used,  or  is  constant  for  constant  values  of  m. 

Another  statement  of  the  law  of  the  power  function  is:  As  x 
increases  in  geometrical  progression,  y,  or  the  power  function,  in- 
creases in  geometrical  progression  also. 

Let  m  be  nearly  1,  say  1  +  r,  where  r  is  the  percent  change  in  x 
and  is  small,  then  we  have: 

y'       fix  +  rx)        a{x  +  rxy        /.    ,     x         -,    ,  /^n 

-  =  - — TT-v—  =  z =  (1  +  r)"  =  1  -h  nr      (5) 

y  f{x)  ax^ 

by  the  approximate  formula  for  the  binomial  theorem  (§111). 

Hence,  re] 

transposing : 


ffx) 
Hence,  replacing   1   on   the  right  side  of    (5)   by  -^-  s    and 


y'  -y     /(^-  +  rx)  -  fix) 

:  — —  = TTT^; =  nr  (o; 

y  fix) 

The  fraction  in  the  first  member  is  the  percent  change  in  y  or  in  fix) . 
The  number  r  is  the  percent  change  in  the  variable  x.  Therefore 
(6)  states  that  for  small  changes  of  the  variable  the  percent  of 
change  in  the  function  is  n  times  the  percent  of  change  in  the  variable. 
Let  the  exponential  function  be  represented  by 

y  =  ae^^  =  Fix)  (7) 

As  already  noted  in  the  preceding  sections,  increasing  x  by  a  con- 
stant term  increases  y  or  the  function  by  a  constant  factor.     Thus 

t_Fix  +  h)^a^i^--^_ 

y   ~      Fix)  ae^-  ^  ^ 

which  is  independent  of  the  value  of  x  or  is  constant  for  constant  h. 
The  expression  e^''  is  the  factor  by  which  y  or  the  function  is  in- 
creased when  X  is  increased  by  the  term  or  increment  h.     See  also 
§147  and  Fig.  97. 
In  other  words,  as  x  increases  in  arithmetical  progression,  y 


§156]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   267 

or  the  exponential  function  increases  in  geometrical  progression. 
The  percent  of  change  is: 

F{x)  e         i  {\)) 

which  is  constant  for  constant  increments  h  added  to  the  variable  x. 

If  X  change  by  a  constant  percent  from  x  to  a;(l  +  r),  it  will  be 
found  that  the  percent  change  in  the  function  is  not  constant,  but 
is  variable. 

The  above  properties  enable  one  to  determine  whether  measure- 
ments taken  in  the  laboratory  can  be  expressed  by  functions  of 
either  of  the  types  discussed;  if  the  numerical  data  satisfy  the 
test  that  if  the  argument  change  by  a  constant  factor  the  function 
also  changes  by  a  constant  factor,  then  the  relation  may  be  repre- 
sented by  a  power  function.  If,  however,  it  is  found  that  a  change 
of  the  argument  by  a  constant  increment  changes  the  function 
by  a  constant  factor,  then  the  relation  can  be  expressed  by  an 
equation  of  the  exponential  type. 

We  have  already  shown  how  to  determine  the  constants  of  the 
exponential  equation  by  graphing  the  data  upon  semi-logarithmic 
paper.  In  case  the  equation  representing  the  futiction  is  of  the 
form: 

y  =  ae^*  +  c  (10) 

then  the  curve  is  not  a  straight  line  upon  semi-logarithmic  paper. 
If  tabulated  observations  satisfy  the  condition  that  the  function 
less  (or  plus)  a  certain  constant  increases  by  a  constant  factor  as 
the  argument  increases  by  a  constant  term,  then  th«  equation  of 
the  type  (10)  represents  the  function  and  the  other  constants  can 
readily  be  determined. 

The  determination  of  the  equations  of  curves  of  the  parabolic 
and  hyperbolic  type  is  best  made  by  plotting  the  observed  data 
upon  logarithmic  coordinate  paper  as  explained  in  the  next 
section. 

156.  Logarithmic  Coordinate  Paper.  If  coordinate  paper  be 
prepared  on  which  the  uniform  x  and  y  scales  are  both  replaced 
by  non-uniform  scales  divided  proportionately  to  log  x  and  log  y 
respectively,  then  it  is  possible  to  show  that  any  curve  of  the  para- 
bolic or  hyperbolic  type  when  drawn  upon  such  coordinate  paper  will 


268        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§156 

h6  a  straight  line.    This  kind  of  squared  paper  is  called  logarithmic 
paper,  and  is  illustrated  in  Fig.  105. 

To  tind  the  equation  of  a  line  PQ  on  such  paper,  we  imagine,  as 
in  the  case  of  semi-logarithmic  paper,  that  all  rulings  are  erased 
and  replaced  by  continuations  of  the  uniform  scales  ON  and  MN, 
on  which  the  length  ON  or  MN  is  taken  as  unity.     Denoting,  as 


oM 


ION 


,, ,  II , , ,  ,1 , , , , , , , ,  ,1 , , 

II 

il 

JM,,,M 

n    uin, 

1 

r   8 

z.  y 

^ 

^ 

" 

X' 

^ 

^ 

^ 

^ 

y^ 

y^' 

^  5 

^ 

^ 

^ 

r    ^ 

'p 

=  2 

L 

r  1 

-^0 

1  2  3  456789     10 

Single  Logarithmic,  Scale  of  Oommon  Logaritlims  in  Margins 

Fig.   105. — Logarithmic  Coordinate  Paper,  Form  ilf4.     The  finer  rulings 
of  form  M 4  are  not  reproduced. 


before,  distances  referred  to  these  uniform  scales  by  capital  letters, 
we  may  write  as  the  general  equation  of  a  straight  line: 

Y  =  mX-\-B  (1) 
In  the  case  of  the  line  PQ,  m  =  0.505,  B  =  0.219,  and  hence 

Y  =  0.505X  +  0.219 

But,  Y  =  log  y,  X  =  log  X,  where  y  and  x  represent  distances 


§156]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   269 

measured  on  the  scales  LM  and  LO  respectively,  and  0.219  = 
log  1.65.    Hence: 

log  y  =  0.505  log  X  +  log  1.65 
or, 

log  y  —  log  1.65  =  0.505  log  x 
This  may  be  written  in  the  form 

log  j^  =  log  xo- 505 


whence 

y 


1.65  ~  ^ 


or, 


y  =  1.65x0-505  (2) 

In  general,  if  B  =  log  b,  we  may  write  the  equation  (1)  in  the 
form 

y  =  hx*^  (3) 

If  the  straight  line  on  logarithmic  paper  passes  through  the 
point  (1,  1)  its  Cartesian  equation  is 

Y  =  mX  (4) 

or,  referred  to  the  logarithmic  scales, 

logy  =  m  log  X 
or, 

y  =  x"».  (5) 

If  the  straight  line  on  logarithmic  paper  passes  through  the  point 
(a,  h)  with  slope  m,  its  equation  referred  to  the  logarithmic  scales  is 


6  -  [  aj 


(6) 


On  logarithmic  paper,  form  Af4,  the  numbers  printed  in  the  lower 
and  in  the  left  margin  refer  to  the  non-uniform  scale  in  the  body 
of  the  paper.  By  calling  the  left-hand  lower  corner  the  point 
(1,  10),  (10,  10),  (10,  1),  (10,  100),  (1,  100)  or  (100,  100),  .  .  ., 
instead  of  (1,  1),  these  numbers  may  be  changed  to  10,  20,  30, 
.    .    .   or  to  100,  200,  300,   .    .    . ,  etc. 

In  the  following  exercises  the  graphs  are  to  be  carefully  con- 
structed upon  logarithmic  paper,  and  the  values  of  the  various 
graduations  and  all  other  necessary  information  indicated  on  the 
paper  in  terms  of  the  proper  concrete  units. 


270        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§156 

If  the  range  of  any  variable  is  to  extend  beyond  any  of  the  single 
decimal  intervals,  1—10,  10—100,  100—1000,  .  .  .,  the 
"multiple  paper,"  form  MQ,  may  be  used,  or  several  straight  lines 
may  be  drawn  across  form  M4  corresponding  to  the  value  of  the 
function  in  each  decimal  interval,  1 — 10,  10 — 100,  .  .  .,  so 
that  as  many  straight  lines  will  be  required  to  represent  the  func- 
tion on  the  first  sheet  as  there  are  intervals  of  the  decimal  scale  to 
be  represented.  However,  if  the  exponent  m  in  y  =  bx"^  he  & 
rational  number,  say  nfr,  then  the  lines  required  for  all  decimal 
intervals  will  reduce  to  r  different  straight  lines. 

One  of  the  most  important  uses  of  logarithmic  paper  is  the  de- 
termination of  the  equation  of  a  curve  satisfied  by  laboratory 
data.  If  such  data,  when  plotted  on  logarithmic  paper,  appear 
as  a  straight  line,  an  equation  of  the  parabolic  type  satisfies  the 
observations  and  the  equation  is  readily  found.  The  exponent 
m  is  determined  by  measuring  the  slope  of  the  line  with  an  ordi- 
nary uniform  scale.  The  equation  of  the  line  is  best  found  by 
noting  the  coordinates  of  any  one  point  (a,  b)  and  substituting 
these  and  the  slope  m  in  the  equation 


b  -  la] 


Exercises 

Draw  the  following  on  single  or  multiple  logarithmic  paper,  forms 
ikf4or  MQ: 

1.  y  =  X,  y  =  2x,  7j  =  Sx,  y  =  4x,   .    .    . 

2.  y  =  X,  y  =  x^,  y  =  x^,  y  =  x^,    .    .    . 

3.  y  =  1/x,  y  =  l/x2,  y  =  l/x',    .    .    . 
4i,  y  =  x^^ ,  y  =  x^,  y  =  x^^,   ... 

5.  A  =  xr2. 

6.  7?  =  0.003y2,  where  p  is  the  pressure  in  pounds  per  square  foot 
on  a  flat  surface  exposed  to  a  wind  velocity  of  v  miles  per  hour. 

7.  V  =  cVrs  for  c  =  110  and  r  =  1. 

8.  /  =  V2^  for  g  =  32.2. 

9.  C  =  E/R  where  ^  =  110  volts. 

10.  s  =  (1/2)^^2  where  g  =  32.2. 

11.  T  =  irVLjg,  where  g  =  32.2. 


§156]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   271 

12.  p/po  =  (p/po)^•"^  where  po  =  0.075,  the  weight  of  1  cubic 
foot  of  air  in  pounds  at  70"  F.  and  at  pressure  po  of  14.7  pounds  per 
square  inch. 

13.  H  =  ^~,  for  D  =  5000,  10,000,  15,000  20,000,  where  C  = 
225,  D  is  displacement  in  tons  and  S  is  speed  in  knots. 

14.  H  =  -^,  for  N  =  100,  200,  300,  400,  500,  600,  700,  800, 

900,  1000.     d  is  the  diameter  of  cold  rolled  shafting  in  inches;  the 
line  "should  be  graphed  for  values  of  d  between  d  =  1  and  d  =  10. 


-k  -'^ 

^^i^' 

:t::t;:^ 

1 
as 

:;:; 

= 

=E 

:±z!::::? 

■^ 'y 

—7 

^  ^^ 

iff 

4iL. 

0.7 

0.C 

;;:; 

^ 

1 

-:"i2::^ 

?i^i^s 

y 

^Vf^U' 

\yr7  xy^. 

r^   '  i   +^     /  '  / 

n  i 

\.'^  ':Sx 

!    i  1  ^' 

Ip^f.-'V: 

'  ^ '. '  3 ' 

as 

=;:: 

1  = 

:ttt^'  j^* 

miUf 

z 

.=:|i:_ 

a2 

. 

_ 

^l 

Mi| 

;:2 

sa£= 

4Mf 

;  +  |+  - 

^  ^ 

'''<<2'' 

^  . ' .  1 

>^=^J 

2 ' 

,  • 

'  ^ 

^ , ' 

,^^g. 

.^..., 

^2?^- 

,'', 

^ 

'y_ ' 

:;^ 

^'  ,'.■-> 

-x^ 

ai 
ao9 
ao8 

'M 

i 

=  : 

^'TT, 

^    J 

ao7 
aoc 

E 

;2l 

y|-:::: 
J  g  __.-  +  +  .. 

1 

-: 

^ 

—  ; 

v^^S 

I        i 

n 

ao4 

^ 

^    / 

^^/^yM\ 

1  :::i:: ,. 

-i 

E!    — , 

,ai 


0.2 


4     5    6   7  8  910 


---«.  —    as    ai     ae  as^i-o  2 

"•"8  3/  as    a7  0.9 

Formula:  q,  =  3.37 iff  ^2       Discharge  over  Trapezoidal  "Wier 
i^Length  of  Crest  in-Feet 
/f  =Head  on  Crest  in  Feet 
q  =  Discharge  in  Second  Feet 

Fig.   106. — A  Weir  Formula  Graphed  on  Multiple  Logarithmic  Paper. 


15.  F  =  0.000341  TFi?iV2,  where  N  is  revolutions  per  minute,  R  is 
radius  in  feet,  W  is  weight  in  pounds,  and  F  is  centrifugal  force  in 
pounds. 

16.  q  =  3.37L/i'/-  for  L  =  0.1,  0.2,  0.3,  0.4,  0.5,  0.6,  0.7,  0.8,  0.9, 
1.0.     See  Fig.  106. 

0.387186 

17.  H  =  — ^^^ — ,  where  V  is  the  velocity  of  water  in  feet  per 

second  under  the  head  of  H  feet  per  10,000  feet  in  clean  cast-iron  pipe 
of  diameter  d  feet.    See  Fig.  108. 

18.  The  relation  between  electrical  resistance  and  amount  of  total 
solids  in  solution  for  Arkansas  River  Valby  water  at  70"  F.  is  given 
by  the  following  table: 


272        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§157 


S  —  total  solids  in  solution  as  parts  per  1,000,000: 

1,000,  800,  600,  400,  300,  200 
R  =  resistance  in  ohms:     215,  260,  340,  480,  615,  860. 

Plot  the  results  on  form  M 4  and  find  from  the  graph  the  equation 
connecting  S  and  R. 


oo  :^ooo < 
ocJooooc 

n  II  11  fl  II  II 


o  o 


150 
140 

y 

m 

71 

130 
120 
110 
100 

f/// 

11 

/ 

/// 

1 

/ 

1 

/// 

'li 

i 

1 

// 

1 

"S  QO 

/ 

h 

7/ 

1 

1 

1  "^^ 

60 

50 

* 

'k 

/ 

I 

f 

1 

1 

<t 

V^ 

Jt 

r 

1 

//// 

'/// 

1 

d 

i 

/// 

// 

40 

If 

'III//, 

'//// 

/ 

on 

h 

m 

/ 

20 

10 

0 
( 

W/ 

¥ 

^ 

W 

1 

# 

) 

] 

L 

2 

5 

1 

6 

d  in  Feet 

Fig.   107. — Capacity  in  Cubic  Feet  per  Second  of  Trapezoidal  Smooth 
Concrete  Flumes  for  Various  Gradients  {S)  and  for  Various  Dimensions  (d) , 

19.  Replot  the  curves  of  Fig.  107.  On  the  new  diagram  draw  the 
lines  corresponding  to  slopes  of  7,  8,  and  9  feet  per  10,000  respectively. 

20.  Explain  the  periodic  character  of  the  rulings  on  Figs.  106 
and  108. 

157.  Stuns  of  Exponential  Functions.  Functions  consisting  of 
the  sum  of  two  different  exponential  functions  are  of  frequent 
occurrence  in  the  application  of  mathematics,  especially  in  elec- 


§157]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   273 

trical  science.  Types  of  fundamental  importance  are  e"  +e~« 
and  e"  —  e-«  which  are  so  important  that  the  forms  (e"  +e-«)  /2 
and  (e«  —  e-")  /2  have  been  given  special  names  and  tables 
of  their  values  have  been  computed  and  printed.     The  first  of 


rriction  Head  in  Feci  per  1000  Ft.of  Pipe 

g     §   S  §         -.        ?J     5  -^  5        <=         "=>,    '=i   °,    °.    <i 


Note: 

For  ojien  conduits,  multiply  UydraTille  Radius  bj  4  to 
get  Equivalent  Diameter.    Diagram  gives  nearly  same 
_QQ    results  as  Kutters  Tormula  with   n  =.011. 

Fur  old  or  foul  pipes  multiply  required  head  by  1.45 
200    to  1-03  or  divide  diagram  velocity  by  1.20  to  1.28  for 
^    T—  2  to  5  feet  per  second .  j 

7,'^J^o     o    o  o         S    For  Buoh  pipes  H=  0.50  ^13 
--""*'"        -100  '' 


Mm4im^^ 


Diagram  of  Flow  In  Clean  Oast  Iron  or  Wrought  Iron  Pipes 
Based  ou  the  Formula,  H,  in  Feet  per  1000  Feet  =  0.33  ji^ 

Fig.  108. — A  Complicated  Example  of  the  Use  of  Multiple  Logarithmic 
Paper,  Form  M6.     From  Transactions  Am.  Soc.  C.  E.  Vol.  LI. 


these  is  called  the  hyperbolic  cosine  of  u  and  the  second  is  called  the 
hjrperbolic  sine  of  u;  they  are  written  in  the  following  notation: 


cosh  w  =  (e"  +  e-«)  /2,  sinh  u  =  (e»  —  e-«)  /2 


18 


274        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§157 

If  X  =  a  cosh  u  and  y  =  a  sinh  u,  then  squaring  and  subtracting 
a;2  _  2/2  =  a^{cosh^  u  —  sinh^  u) 


...[.i±-t 


e2.  _  2  4-  e- 


Therefore  the  hyperbolic  functions 

X  =  a  cosh  u,  y  =  a  sinh  w 


4 

J 

' 

, 

3.5 

y-- 

'-^ 

V 

3 

•\ 

I 

\ 

2.5 

/ 

^ 

\ 

2 

>/ 

\ 

1,5 

f 

^ 

^1 

J 

i 

^ 

^  = 

2/ 

=  h 

ga 

rA 

^ 

y 

v 

1/ 

?}^e'- 

c 



J- 

^ 

p/ 

^ 

-3.5 

-a 

-it 

'■-2 

-U 

-1 

- 

H 

0 

5 

1 

1.5 

2 

^5 

3 

3.5 

/ 

/ 

1.5 

/ 

-2 

/ 

_ 

2.5 

/ 

-3 

1 

. 

3.5 

/ 

-4 

/ 

. 

4.5 

-5 

Fig.   109. — The  Curves  of  the  Hyperbolic  Sine  and  Cosine. 


appear  in  the  parametric  equations  of  a  rectangular  hyperbola 
a;2  _  2/2  =  ^2^ 

just  as  the  circular  functions 

X  =  a  cos  6,  y  =  a  sin  6 
appear  in  the  parametric  equations  of  the  circle 

X^  -{-  yi  =  a^ 

The  graphs  oi  y  =  a  cosh  x  and  y  =  a  sinh  x  were  called  for  in 
exercises  1,  2,  §146.  They  are  shown  in  Fig.  109.  The  first 
of  these  curves  is  formed  when  a  chain  is  suspended  between  two 
points  of  support;  it  is  called  the  catenary.     These  two^curves 


§157]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   275 


are  best  drawn  by  averaging  the  ordinates  of  2/  =  e*  and  y  =  e~*, 
and  the  ordinates  of  i/  =  e»  and  y  =  —  e~'. 

Curves  whose  equations  are  of  the  form  y  =  ae*"*  -(-  6e»*  take 
on  quite  a  variety  of  forms  for  various  values  of  the  constants.  A 
good  idea  of  certain  important  types  can  be  had  by  a  comparison  of 
the  curves  of  Fig.  110  whose  equations  are: 


1  75 

1  ') 

1  ?'i 

\(1) 

(l)^.e"*+  0.5e"2* 
(2)  l/-e-='+o.2e-2« 

(4)2/-e-''.  0.2e"2^ 
(5)2/ee-*.  0.5e"2« 
(6)i/=e-*-  0.80-2'^ 
(7)j/-e-'^_g-2x 
(8)2/.e-*-  1.5e-2» 

1 

.75 

IvO 

5 

,^ 

25 

(6)^ 

W 

r. 

^  ^ 

" — 

0 
-.25 

-.5 

f 

5 

I 

1.6 

2 

2.5 

1 

Fig.   110. — Combinations  of  Two  Exponential  Curves.     After  Steinmetz. 

1/  =  e-*  +  0.5e-2' 
2/  =  e-*  -I-  0.2e-2* 
y  =  e-' 

y  =  e-^  -  0.2e-2^ 
?/  =  e-*  —  0.5e~2* 
2/  =  e-x  _  0.8e-2* 
2/  =  e-'  —  e-2x 
y  =  e-'^  —  1.5e-2-= 
The  student  should  arrange  in  tabular  form  the  necessary 
numerical  work  for  the  construction  of  these  curves. 


276 


ELEMENTARY  MATHEMATICAL  ANALYSIS 


:§i58 


If  the  second  exponent  be  increased  in  absolute  value,  the  points 
of  intersection  with  the  y-Sixis  remain  the  same,  but  the  region  of 
close  approach  of  the  curves  to  each  other  is  moved  along  the  curve 
y  =  e-*  to  a  point  much  nearer  the  ?/-axis.  To  show  this  the  fol- 
lowing curves  have  been  drawn  and  shown  in  Fig.  111. 


IT') 

15 

1.25 
1 

,(1) 

U2) 

(1)  2/=e"^+0.5e'^°* 
(2)?/=^-^ 
C3)2/=6-^-  O.le-10^ 

75 

5 

F 

\, 

?'i 

hB) 

\ 

\ 

^ 

^ 

0 
.25 

.5 

5 

1 

1.5 

2 

2.5 

Fig.   111. — Combinations  of  Two  Exponential  Curves.     After  Steinmetz. 


y  = 

e-^ 

+  0.5e- 

-lOx 

y  = 

6"=^ 

y  = 

g-x 

-  O.le- 

-lOx 

y  = 

Q-X 

-  0.5e- 

-10a: 

y  = 

e-* 

-  e-io^ 

y  = 

e-* 

-  1.5e- 

■10a: 

158.  *  Damped  Vibrations.  If  a  body  vibrates  in  a  medium  like 
a  gas  or  liquid,  the  amplitude  of  the  swings  are  found  to  get  smaller 
and  smaller,  or  the  motion  slowly  (or  rapidly  in  some  cases)  dies 
out.     In  the  case  of  a  pendulum  vibrating  in  oil,  the  rate  of 


§158]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS  277 

decay  of  the  amplitude  of  the  swings  is  rapid,  but  the  ordinary- 
rate  of  the  decay  of  such  vibrations  in  air  is  quite  slow.  The  ratio 
between  the  lengths  of  the  successive  amplitudes  of  vibration  is 
called  the  damping  factor  or  the  modulus  of  decay. 

The  same  fact  is  noted  in  case  the  vibrations  are  the  torsional 
vibrations  of  a  body  suspended  by  a  fine  wire  or  thread.  Thus  a 
viscometer,  an  instrument  used  for  determining  the  viscosity  of 
lubricating  oils,  provides  means  for  determining  the  rate  of  the 
decay  of  the  torsional  vibration  of  a  disk,  or  of  a  circular  cylinder 


0*5 

/ 

^ 

V 

/ 

\ 

f 

\ 

<^ 

/ 

r 

JTTI 

0     11     12  47r 

t 

U 

'       f 

V 

: 

'A 

r   ' 

r     { 

r^ 

^f 

0.5 

\ 

^^ 

/ 

Fig.  112.— The  Curve  y  =  e"'  '^  sin  t. 


suspended  in  the  oil  by  a  fine  wire.  The  "amplitude  of  swing" 
is  in  this  case  the  angle  through  which  the  disk  or  cylinder  turns, 
measured  from  its  neutral  position  to  the  end  of  each  swing. 

In  all  such  cases  it  is  found  that  the  logarithms  of  the  successive 
amplitudes  of  the  swings  di^er  by  a  certain  constant  amount  or,  as 
it  is  said,  the  logarithmic  decrement  is  constant.  Therefore  the 
amplitudes  must  satisfy  an  equation  of  the  form 

A  =  ae~^' 

where  A  is  ampUtude  and  t  is  time.  The  actual  motion  is  jgiven 
by  an  equation  of  the  form 

y  =  ae-*<  sin  ct 

A  study  of  oscillations  of  this  type  will  be  more  fully  taken  up  in 


278        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§158 

the  calculus,  for  the  present  it  will  suffice  to  graph  a  few  examples 
of  this  type.    Let  the  expression  be 

y  =  e-</5  sin  t  (1) 

A  table  of  values  of  t  and  y  must  first  be  derived.  There  are  three 
ways  of  proceeding:  (1)  Assign  successive  values  to  t  irrespective 
of  the  period  of  the  sine  (see  Table  V  and  Fig.  112).  (2)  Select 
for  the  values  of  t  those  values  that  give  aliquot  parts  of  the  period 
27r  of  the  sine  (see  Table  VI  and  Fig.,  113).  (3)  Draw  the  sinu- 
soid y  =  sint  carefully  to  scale  by  the  method  of  §55;  then  draw 
upon  the  same  coordinate  axes,  using  the  same  units  of  measure 

1.5 

y 
1 


■1.6 


\ 

V 

/ 

N 

i  ^^ 

,^* 

' 

/ 

\ 

^"^ 

-^ 

^^ 

/ 

\ 

IT 

27r 

^ 

"^ 

"sT 

IT" 

AT 

2       i 

:             ( 

\     8      1 

1°/ 

2     1 

4      1 

6     llN^ 

L-2 

^.^9,A       9R       O 

8 

\ 

\ 

^ 

^ 

J 

^ 

Fig.   113.— The  Curve  y  =  e-'/5  gin  t. 


adopted  for  the  sinusoid,  the  exponential  curve  y  =  e-'  Z^;  finally 
multiply  together,  on  the  slide  rule,  corresponding  ordinates  taken 
from  the  two  curves,  and  locate  the  points  thus  determined. 

The  first  method  involves  very  much  more  work  than  the  second 
for  two  principal  reasons:  First,  tables  of  the  logarithms  of  the 
trigonometric  functions  with  the  radian  and  the  decimal  divisions 
of  the  radian  as  argument  are  not  available;  for  this  reason  57.3° 
must  be  multiplied  by  the  value  of  t  in  each  case  so  that  an  ordinary 
trigonometric  table  may  be  used;  second,  each  of  the  values  written 


§158]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   279 


in  column  (3)  of  the  table  must  be  separately  determined,  while 
if  the  periodic  character  of  the  sine  be  taken  advantage  of,  the 
numerical  values  would  be  the  same  in  each  quadrant. 

The  second  method,  because  of  the  use  of  aliquot  divisions  of 
the  period  of  the  sine,  such  as  7r/6  or  7r/12  or  7r/18  or  7r/20,  etc., 
possesses  the  advantage  that  the  values  used  in  column  (3)  need 
be  found  for  one  quadrant  only  and  the  values  required  in  column 
(2)  are  quite  as  readily  found  on  the  slide  rule  as  in  the  first 
method. 

TABLE  V 


Table  of  the  function  y 

=  e-'/s  sin  t 

1    1 

2 

3 

4 

5 

t  in  radians  I 

log  e-'/*  = 
-  (0.0869X 

log  sin  t  or  log 

sin  57.3t  if  t  is 

in  degrees 

log  J/ 

y 

0.0  ! 

-  0.0000 

+  0.000 

0.5 

-  0.0434 



9.6807 

9.6372 

+  0.434 

1.0 

-0.0869 

9.9250 

9.8381 

+  0.689 

1.5 

-  0.1303 

9.9989' 

9.8686 

+  0.739 

2.0 

-  0.1737 

9.9587 

9 . 7850 

+  0.610 

2.5 

-  0.2172 

9.7771 

9.5599 

+  0.363 

3.0 

-  0.2606 

9.1498 

8.8892 

+  0.077 

3.5 

-  0.3040 

9.5450 

9.2410 

-  0.174 

4.0 

-  0.3474 

9.8790 

9.5312 

-  0.340 

4.5 

-  0.3909 

9.9901 

9.5992 

-  0.397 

5.0 

-  0.4343 

9.9818 

9.5475 

-0.353 

5.5 

-  0.4777 

9.8485 

9.3708 

-  0.235 

6.0 

-  0.5212 

9.4464 

8.9252 

-  0.084 

6.5 

-  0.5646 

9.3322 

8.7679 

+  0.059 

7.0 

-  0.6080 

9.8175 

9.2095 

+  0.162 

7.5 

-  0.6515 

9.9722 

9.3207 

+  0.209 

8.0 

-  0.6949 

9.9954 

9 . 3005 

+  0.200 

8.5 

-  0.7383 

9.9022 

9.1634 

+  0.146 

9.0 

-0.7817 

9.6149 

8.8332 

+  0.068 

9.5 

-  0.8252 

8.8760 

8.0508 

-0.011 

10.0 

-  0.8686 

9 . 7356 

8.8670 

-0.074 

10.5 

-  0.9120 

9.9443 

9.0323 

-0.108 

11.0 

-  0.9555 

9.9999 

9.0444 

-  0.111 

11.5 

-  0.9989 

9.9422 

8.9433 

-0.088 

12.0 

-  1.0424 

9.7296 

8.6872 

-0.049 

280        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§158 


TABLE  VI 
Table  of  the  function  y  =  e"'/^  sin  t 


1 

2 

3 

4 

5 

n  =  t  in 

units  of 

7r/6  radians 

loge-nV"  = 
-  (0.0455)n 

iog.sin  nx/6 

logy 

y 

0 

1 

-  0.0000 

-  0.0455 

0.000 
+  0.450 

9.6990 

9.6535 

2 

-  0.0910 

9.9375 

.   9.8465 

+  0.702 

3 

-  0.1364 

0.0000 

9.8636 

+  0.731 

4 

-  0.1819 

9.9375 

9.7556 

+  0.570 

5 

-  0.2274 

9.6990 

9.4716 

+  0.296 

6 

-  0.2729 

+  0.000 

7 

-  0.3184 

9.6990 

9 . 3806 

-  0.240 

8 

-  0.3638 

9.9375 

9.5737 

-0.375 

9 

-0.4093 

0.0000 

9.5907 

-  0.390 

10 

-  0.4548 

9.9375 

9.4827 

-0.304 

11 

-0.5003 

9.6990 

9.1987 

-  0.158 

12 
13 

-  0.5458 
-0.5912 

0.000 
+  0.128 

9.6990 

9.1078 

14 

-  0.6367 

9.9375 

9.3008 

+  0.200 

15 

-  0.6822 

0.0000 

9.3178 

+  0.208 

16 

-0.7277 

9.9375 

9.2098 

+  0.162 

17 

-0.7732 

9.6990 

8.9258 

+  0.084 

18 
19 

-0.8186 
-0.8641 

0.000 
-  0.068 

.   9.6990 

8.8349 

20 

-  0.9016 

9.9375 

9.0279 

-  0.107 

21 

-  0.9551 

0.0000 

9 . 0449 

-0.111 

22 

-  1.0006 

9.9375 

8.9369 

-0.087 

23 

-  1.0460 

9.6990 

8.6530 

-  0.045 

24 

-  1.0915 

0.000 

The  third  method  is  perhaps  more  desirable  than  either  of  the 
others  if  more  than  two  figures  accuracy  is  not  required.  The 
curve  can  readily  be  drawn  with  the  scale  units  the  same  in 
both  dimensions,  as  is  sometimes  highly  desirable  in  scientific 
applications. 

In  Figs.  112  and  113  a  larger  unit  has  been  used  on  the  vertical 
scale  than  on  the  horizontal  scale.  In  Fig.  113  the  horizontal  unit 
is  incommensurable  with  the  vertical  unit.  To  draw  the  curve 
to  a  true  scale  in  both  dimensions  it  is  preferable  to  lay  off  the 


§158]  LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS   281 

coordinates  on  plain  drawing  paper  and  not  on  ordinary  squared 
paper.  Rectangular  coordinate  paper  is  not  adapted  to  the  proper 
construction  and  discussion  of  the  sinusoid,  or  of  curves,  like  the 
present  one,  that  are  derived  therefrom. 

Curves  whose  equations  are  of  the  form  y  =  ^e~'/^  sin  t  or 
y  =  3e~'/^  sin  t,  etc.,  are  readily  constructed,  since  the  constants 
1/2,  3,  etc.,  merely  multiply  the  ordinates  of  (1)  by  1/2,  3, 
etc.,  as  the  case  may  be.  Likewise  the  curve  y  =  e~^*  sin  ex  is 
readily  drawn  since  sin  ex  can  be  made  from  sin  x  by  multi- 
plying all  abscissas  of  sin  a:  by  1  /c. 


CHAPTER  IX 


TRIGONOMETRIC    EQUATIONS    AND    THE    SOLUTION     OF 
TRIANGLES 

A.  FURTHER  TRIGONOMETRIC  IDENTITIES 

159.  Proof  that  p  =  a  cos  0  +  b  sin  ^  is  a  Circle.  I.  Geomet- 
rical Explanation.  We  know  (§64)  that  pi  =  a  cos  6  is  the  polar 
equation  of  a  circle  of  diameter  a,  the  diameter  coinciding  in 
direction  with  the  polar  axis  OX;  for  example,  the  circle  OA^ 

Fig.  114.  Likewise,  p2  = 
b  sin  ^  is  a  circle  whose  dia- 
meter is  of  length  b  and 
makes  an  angle  of  +  90° 
with  the  polar  axis  OX,  as 
the  circle  OB,  Fig.  114. 
Also,  p  =  c  cos  {6  —  6\)  is 
a  circle  whose  diameter  c 
has  the  direction  angle  9i. 
See  equation  (4),  §68.  We 
shall  show  that  if  the  radii 
vectores  corresponding  to 
any  yalue  of  B  in  the  equa- 
tions pi  —  a  cos  6  and  p2  = 

b  sin  B  be  added  together  to 
Fig.   114.-Combination    of    the    Cir-  ^^^^    ^   ^^^   ^.^^i^g    ^e^^Oj. 
cles  p  =  a  cos  6  and  p  =  b  sm  6  into  a 

Single  Circle  p  =  a  cos  6  +  b  sin  6.  P,  then,  for  all   values   of  B, 

the  extremity  of  p  lies  on 
a  circle  (the  circle  00,  Fig.  114)  of  diameter  \/a^  +  b^.  In 
other  words  we  shall  show  that: 

p  =  a  cos  ^  +  6  sin  ^  (1) 

is  the  equation  of  a  circle. 

282 


§159]  TRIGONOMETRIC  EQUATIONS  283 

In  Fig.  114,  pi  =  a  cos  d  will  be  called  the  a-circle  OA;  pa  = 
b  sin  0  will  be  called  the  b-circle  OB.  For  any  value  of  the  angle 
6  draw  radii  vectores  OM,  ON,  meeting  the  a-  and  6-circles 
respectively  at  the  points  M  and  N.  If  P  be  the  point  of  inter- 
section of  MN  produced  with  the  circle  whose  diameter  is  the 
diagonal  OC  of  the  rectangle  described  on  OA  and  OB,  we  shall 
show  that  OM  +  ON  =  OP,  no  matter  in  what  direction  OP  be 
drawn. 

Let  the  circle  last  mentioned  be  drawn,  and  project  BC  on  OP. 
Since  ONB  and  OPC  are  right  angles,  NP  is  the  projection  of 
BC  {=  a)  upon  OP.  But  OM  also  is  the  projection  of  a  (=  OA) 
upon  OP.  Hence  NP  =  OM  because  the  projections  of  equal 
parallel  lines  on  the  same  line  are  equal.  Therefore,  for  all  values 
of  d,  NP  =  pi  and  OP  =  ON -\- NP  =  p2 -\-  pi,  which  is  the  fact 
that  was  to  be  proved. 

Designating  the  angle  AOC  by  di,  the  equation  of  the  circle  OC  is 
by  §68 

p.  =  Va2  4-  b^  cos  (d  -  ^i)  (2) 

The  value  of  6i  is  known,  for  its  tangent  is  -.    It  should  be  observed 

that  there  is  no  restriction  on  the  value  of  0.  As  the  point  P 
moves  on  the  circle  OC,  the  circumference  is  twice  described  as  6 
varies  from  0°  to  360°,  but  the  diagram  for  other  positions  of  the 
point  P  is  in  no  case  essentially  different  from  Fig.  114. 

The  above  reasoning  and  the  diagram  involve  the  restriction 
that  both  a  and  b  are  positive  numbers.  While  it  is  possible  to 
supplement  the  reasoning  to  cover  the  cases  in  which  this  restric- 
tion is  removed,  it  will  be  unnecessary  as  the  analytical  proof  at 
the  end  of  this  section  is  applicable  for  all  values  of  a  and  b. 

Example:  From  the  above  we  know  that  the  equation 
p  =  6  cos  ^  +  8  sin  0  is  a  circle.  The  diameter  of  the  circle  is 
\^a^  -h  62  =  V  6^  +  8^  =  10,  so  that  the  equation  of  the  circle 
may  also  be  written  in  the  form  p  =  10  cos  {0  —  di),  in  which  ^i 

is  the  angle  whose  tangent  is  -  =  «  =  1.33.    From  a  table  of  tan- 

fl         o 

gents  di  =  53°  8',  so  that  the  equation  of  the  circle  may  be 
written  p  =  10  cos  {0  -  53°  8'). 

//.  Analytical  Proof.     We  shall  prove  analytically  that  p  = 


284        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§159 

a  cos  0  +  6  sin  0  is  a  circle,  without  imposing  conditions  upon  the 
algebraic  signs  of  a  and  h.     Multiply  both  members  of 

p  =  a  cos  6  -^  bsin  6  (4) 

by  p,  and  obtain 

p2  =  ap  cos  6  -\-  bp  sin  d  (5) 

By  §70,  the  expression  p  cos  6  is  the  value  in  polar  co- 
ordinates of  the  Cartesian  abscissa  x;  also  p  sin  d  is  the  value  in 
polar  coordinates  of  the  ordinate  y.  Likewise  p^  =  x"^  +  y^. 
After  the  substitution  of  these  values,  (5)  becomes 

x^-\-y'^  =  ax  +  by  (6) 

Transposing  and  completing  the  squares: 

r       ay  ,  r       by     a^  +  b^ 

This  is  the  Cartesian  equation  of  a  circle  with  center  at  the  point 

\^y  2  r  ^^d  of  radius  ^\/a^  +  b^.     The  circle  passes  through 

the  origin,  since  the  coordinates  (0,  0)  satisfy  the  equation,  and 
also  passes  through  the  point  (a,  b)  since  these  coordinates 
satisfy  (6). 

Since  (4)  is  now  known  to  represent  a  circle  passing  through  the 
origin,  its  polar  equation  can  be  written  in  any  of  the  forms  (3)- 
(6)  of  §68.  Calling  di  the  direction  angle  of  the  diameter 
of  (7),  (no  matter  what  direction  OC  actually  occupies)  we  can 
write  the  eqation  of  the  circle  in  the  form 

P  =  \/a2  +  fe2  cos  {d  -  di)  (8) 

in  which  the  direction  angle  61  is  the  angle  AOC,  Fig.  114,  or  the 
angle  whose  tangent  is  6  -^  a.  If  a  and  b  are  not  both  positive, 
the  angle  di  is  still  easily  determined.  For  example,  if  a  =  —  1, 
and  b  =  —  If  then  61  =  angle  of  third  quadrant  whose  tangent 
is  1,  or  =  225°,  so  that  equation  (8)  becomes: 

p  =  V2C0S  (6  -225°) 
This  may  also  be  written 

p  =  \/2  cos  (^  +  135°) 
since  the  resulting  circle  may  be  thought  of  as  p  =  \/2  cos  6 
rotated  negatively  through  135°. 


§159]  TRIGONOMETRIC  EQUATIONS  285 

The  equation  of  the  circle  OC  in  any  position,  that  is,  for  any 
values  of  a  and  h,  positive  or  negative,  may  also  be  written  in 
the  form 

P  =  Va'  +  h  2  sin  ((9  +  ^2)  (9) ' 

in  which  Bi  is  the  angle  BOC  in  Fig.  114.     See   §68,  equations 
(5)  and  (6). 

It  has  been  emphasized  above  that  6,  Oi,  O2,  are  any  angles — 
that  is,  angles  not  restricted  in  size  or  sign.  The  distinction  be- 
tween them  need  not  be  lost  sight  of,  however.  0  is  any  angle  be- 
cause it  is  the  variable  vectorial  angle  of  any  point  of  the  locus,  and 
ranges  positively  and  negatively  from  0°  to  any  value  we  please. 
Oi  is  anj  angle  (positive  or  negative)  because  it  is  the  direction 
angle  of  the  diameter  OC.  It  is  a  constant,  but  a  general,  or 
unrestricted,  angle,  but  would  usually  be  taken  less  than  360° 
in  absolute  value.  By  construction,  62  is  also  any  constant 
angle. 

The  result  of  this  section  should  also  be  interpreted  when  the 
variables  are  x  and  y  in  rectangular  coordinates,  and  not  p  and 
6  of  polar  coordinates.  Thus,  y  =  a  cos  a;  is  a  sinusoid  with 
highest  point  or  crest  at  a:  =  0,  27r,  47r,   .    .    .     Likewise,  y  = 

b  sin  X  is  a  sinusoid  with  crest  at  x  =  ^'  -^'  -w~'   •    •    •     The 
above  demonstration  shows  that  the  curve 
y  =  a  cos  X  +  ^  sin  re 
is  identical  with  the  sinusoid 


y  =  Va^  +  b^  cos  (x  —  hi)  =  VaM-^  sin  {x  +  ^2) 
of  amplitude  \/a^  +  b^  and  with  the  crest  located  at  x  =  hi,  or  at 

X  —  A2,  where  hi  is,  in  radians,  the  angle  whose  tangent  is  -,  and 
^  a 

h2  is,  in  radians,  the  angle  whose  tangent  is  r* 

Exercises 

1.  Put  the  equation  p  =  2  cos  d  +  2  a/3  sin  6  in  the  form 
{x  —  hy  •\-  {y  —  hy  =  h^  -\-  fc*;  also  in  the  form»p  =  a  cos  {d  —  d\) 
and  find  the  value  of  Oi.     See  equation  (7)  above. 

2.  Find  the  value  of  0i  if  p  =  cos  d  —  \/3  sin  6. 


286        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§160 

3.  Put  the  equation  p  =  4  cos  6  +  ^y/S  sin  6  in  the  forms 
(x  -  hy  +  (y  -  ky  =  r-  +  /c^  and  p  =  o  cos  {6  -  di). 

4.  Put    the    equation  p  =  —  4   cos  0—4    sin  0    in    the    form 
'(x  -  hy  +  (y  -  ky  =  h'-  -\-k^  and  find  the  value  of  02  when   the 

given  equation  is  written  in  the  form  p  =  a  sin  {6  +  62). 

6.  Put  the  equation  p  =  2\/3  cos  d  +  2  sin  6  in  the  form 
(x  —  h^)  -\-  {y  —  k)"'  =  h^  +  fc2;  also  in  the  form  p  =  a  cos  (6  —  61). 

6.  Put  the  equation  p  =  3  cos  0  +  4  sin  d  in  the  form  p  = 
a  sin  (d  +  ^2).  Put  the  same  equation  in  the  form  p  =  a  cos 
{d  -  di).  {di  is  the  angle  AOC,  Fig.  114. 

7.  Put  the  equation  p  =  5  cos  0  +  12  sin  0  in  the  form  p  = 
a  sin  {d  +  ^2);  also  in  the  form  p  =  a  cos  (6  —  di). 

8.  Put  p"=  3  cos  0  +  4  sin  6  in  the  form  {x  -  hy  +  {y  -  ky  = 
h^  4-  k\ 

9.  Put  p  =  5  cos  0  +  12  sin  d  in  the  form  (x  -  hy  +  {y  -  ky  = 
h^  +  k\ 

10.  Put  the  equation  {x  -  ly  +  {y  -  ly  =  2  in  the  form  p  = 
a  sin  {6  +  a)  and  determine  a  and  a. 

11.  Put  the  equation  (x  +  1)2  +  (1/  -  ^3)^  =  4  in  the  form 
p  =  a  sin  (0  —  or)  and  determine  a  and  a. 

12.  Put  the  equation  (x  +  ly  +  (y  -\-  ^Sy  =  4  in  the  form  p  =  " 
a  sin  (d  —  a)  and  determine  a  and  a. 

13.  Put  the  equation  {x  +  1)^  -\-  (y  -\-  ly  =  2  in  the  form  p  = 
a  cos  (0  +  a)  and  determine  a  and  a. 

14.  Put  the  equation  (x  +  ly  +  (y  +  ^Sy  =  4.  in  the  form  p  = 
a  cos  (6  +  a)  and  determine  a  and  a. 

15.  Find  the  maximum  value  of  cos  0  —  Vs  sin  0,  and  determine 
the  value  of  6  for  which  the  expression  is  a  maximum. 

Suggestion:  Call  the  expression  p.  The  maximum  value  of  p  is  the 
diameter  of  the  circle  p  =  cos  6  —  V 3  sin  0.  The  direction  cosine  of 
the  diameter  is  the  value  of  a  when  the  equation  is  put  in  the  form 
p  =  a  cos  (6  —  a). 

16.  Find  the  value  of  d  that  renders  p  =  W3  cos  0  —  5  sin  0  a 
maximum  and  determine  the  maximum  value  of  p. 

17.  Find  the  maximum  value  of  3  cos  t  +  4  sin  t. 

160.  Addition  Formulas  for  the  Sine  and  Cosine.  From  the 
preceding  section,  equations  ( 1 ) ,  (8)  and  (9),  we  know  that  the  equ  a- 


11601 


TRIGONOMETRIC  EQUATIONS 


287 


tion  of  the  circle  OC,  Fig.  115,  may  be  written  in  any  one  of  the 

forms : 

p  =  a  cos  d  -\-  bsin  6  (1) 

p  =  csin  (6  -  02)  (2) 

p  =  ccos  {e  -  Oi)  (3) 

Hence,  for  all  values  of  0,  0\,  and  02, 


sin  {0  —  ^2)  =  -  cos  0  H —  sin  0 
c  c 


cos  {0 


0A  =      cos  ^  +  "  sm 
^       c  c 


(4) 


(5) 


In  each  of  these  equations  c  =  \/a^  +  6^.  The  letters  a  and  b 
stand  for  the  coordinates  of  C  irrespective  of  their  signs  or  of 
the  position  of  C. 


Fig.  115. — The  Circle  p  =  c  cos  (0—  di)  or  p  =  sin  (d  —  $2,)  used  in 
the  Proof  of  the  Addition  Formulas.  Note  that  di  =  90°  +  $2  which  is 
also  true  for  negative  angles,    namely  5i  =  90°  +  82 

Since  (4)  and  (5)  are  true  for  all  values  of  0,  they  are  true  when 
0  =  0°  and  when  0  =  90°. 

First,  let  0  =  0°  in  (4)  and  (5). 

then  from  (4):   a/c  =  sin  (-  0-^  =  -  sin  02  by  §58  (6) 

From  (5):       a/c  =  cos  (-  0i)  =  cos  0i        by  §58  (7) 


288        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§160 

Second,    let    6  =  90°  in  (4)  and  (5). 

then  from  (4):  6 /c  =  sin  (90°  -  ^2)  =  cos  $2  (8) 

From  (5):       b/c  =  cos  (90°  -  ^1)  =  sin  di  (9) 

Substituting  (6)  and  (8)  in  (4);  also  (7)  and  (9)  in  (5),  we  have 

sin  (6  —  62)  =  sin  6  cos  62  —  cos  d  sin  62  (10) 

cos  {6  —  di)  =  cos  6  cos  ^1  +  sin  d  sin  di  (11) 

Since  these  are  true  for  all  values  of  61  and  62,  put  di  =  (—€1) 
and  02  =  (-eg).    Then  by  §58,  (10)  and  (11)  become 

sin  (6  +  €2)   =  sin  6  cos  €2  +  cos  6  sin  €2  (12) 

cos  {d  +  ei)  =  cos  6  cos  ei  —  sin  6  sin  ei  (13) 

To  aid  in  committing  these  four  important  formulas  to  memory, 
it  is  best  to  designate  in  each  case  the  angles  by  a  and  j8,  and 
write  (12)  and  (13)  in  the  form 

sin  (a  +  j9)  =  sin  a  cos  ^  +  cos  a  sin  /3  (14) 

cos  {a  +  p)  =  cos  a  cos  jS  —  sin  a  sin  jS  (15) 
and  also  write  (10)  and  (11)  in  the  form 

sin  (a  —  jS)  =  sin  a  cos  /3  —  cos  a  sin  jS  (16) 

cos  (a  —  j8)  =  cos  a  cos  /3  +  sin  o:  sin  j8  (17) 

The  four  formulas  (14),  (15),  (16)  and  (17)  must  be  committed  to 
memory.  They  are  called  the  addition  formulas  for  the  sine  and 
cosine.  The  above  demonstration  shows  that  the  addition 
formulas  are  true  for  all  values  of  a  and  /5. 

By  the  above  formulas  it  is  possible  to  compute  the  sine  and  cosine 
of  75°  and  15°  from  the  following  data: 

sin  30°  =  1/2  sin  45°  =  W2 

cos  30°  =  Ws  cos  45°  =  W2 

Thus: 

sin  75°  =  sin  (30°  +  45°)  =  sin  30°  cos  45°  +  cos  30°  sin  45° 
=  iW2  +  W3-W2 
=  iV2(V3  +  l) 
Likewise: 

sin  15°  =  sin  (45°  -  30°)   =  jVi  (V3  -  1) 


§161]  TRIGONOMETRIC  EQUATIONS  289 

161.  Addition    Formtila    for    the    Tangent.     Dividing     the 

members  of  (14)  §160  by  the  members  of  (15)  we  obtain: 

,      ,    „       sin  (a  +  /3)      sin  a  cos  /3  +  cos  a  sin  j3     ,,. 

tan  (a  +  p)  = } j — ^  = -. ; — -     (1) 

^         '^^       cos  (a  4-  /3)      cos  a  cos  /3  —  sm  asm  0     ^  ' 

Dividing  numerator  and  denominator  of  the  last  fraction    by 
cos  a  cos  jS 

sin  a  cos  /3      I      cos  a  sin  /3 

cos  a  COS  /3  COS  a  COS  j8  ^^v 

tan  (a  +  /3)  =  cos  or  cos  ^     _   liTTshT^  ^^^ 

COS  a  COS  /3  COS  a  COS  /3 

or: 

/      ■    ON        tan  a  +  tan  /3 

tan  (a  +  jS)  =  ., t — —^ — ^  (3) 

^         ^^       1  —  tan  a  tan  j8  ^^ 

Likewise  it  can  be  shown  ;rom  (16)  and  (17),  §160,  that: 

.       .  „         tan  a  -  tan  /3  .^. 

tan  (a  —  j8)  =  t  "t^z i — ^  (4) 

^         '^'      1  4-  tan  a  tan  /3  ^  '' 

Equations  (3)  and  (4)  are  the  addition  formulas  for  the  tangent. 

Exercises 

1.  Compute  cos  75°  and  cos  15". 

2.  Compute  tan  75°  and  tan  15°. 

3.  Write  in  simple  form  the  equation  of  the  circle 

p  =  sin  0  +  cos  6. 

4.  Put  the  equation  of  the  circle  p  =  3  sin  6  -\-  4  cos  6  in  the  form 
p  =  c  sin  (0  +  ^i)  and  find  from  the  tables,  or  by  the  slide  rule,  the 
value  of  6i. 

5.  Derive  a  formula  for  cot  (o:  +  /3). 

6.  Prove  cos  (s  +  t)  cos  {s  —  t)  =  cos^  s  —  sin^  t. 

7.  Express  in  the  form  c  cos  (a  —  h)  the  binomial  3  cos  a  + 
4  sin  a. 

8.  Express  in  the  form  c  sin  (a  +  b)  the  binomial  5  cos  a  +  12  sin  a. 

9.  Find  the  coordinates  of  the  maximum  point  or  crest  of  the  sinu- 
soid y  =  sin  X  +  V3  cos  x.  [First  reduce  the  equation  to  the  form 
y  =  cs'm  (x  +  a)]. 

10.  Prove  the  addition  foimulas  in  the  following  manner:     (1) 

In  cos  (6  —  di)  =  -  cos  ^  +  -  sin  6,  show  that  a/c  =  cos  di,  b/c  = 
c  c 

19 


290        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§162 


sin  01,  for  all  values  of  di.  (2)  Find  cos  (0  +  ^2)  by  replacing  di 
by  (—^2).  (3)  Find  sin  (0+^1)  by  the  substitution  in  (1)  of 
d  =  (7r/2  —  0).     (4)  Find  sin  (6  -  d^)  by  replacing  0i  by  (-  ^3). 

162.  Functions  of  Composite  Angles.  The  sine,  cosine,  or 
tangent  of  the  angles  (90°  -  d),  (90°  +  S),  (180°  -  6),  (180°  -(-  6), 
(270°  -  6),  (270°  +  6)  can  be  expressed  in  terms  of  functions 
of  B  alone  by  means  of  the  addition  formulas  of  §§160  and  161. 
If  6  be  an  angle  of  the  first  quadrant,  it  is  easy,  however,  to  obtain 
all  the  relations  by  drawing  the  triangles  of  reference  for  the 
various  angles  and  then  comparing  homologous  sides  of  the  similar 


(-fc.h)  P2 


P,(k.h) 


(h.  k ) 


P(h.k) 


PA-h.-k) 


ih,'k) 


i-k.-h) 


Pi  (k..h) 


A  B 

Fig.   116. — An  Angle  6  Combined  with  an  Even  Number  of  Right  Angles, 
(4)  and  with  an  Odd  Number  of  Right  Angles,  {B). 

right  triangles  of  reference.  Let  the  terminal  side  of  the  angle  B 
be  OP  (Fig.  116  5),  and  let  P  be  the  point  {h,  k).  Let  the 
terminal  sides  of  the  angles  (90°  -  B),  (90°  +  0),  (270°  -  B),  etc., 
be  cut  by  the  circle  of  radius  a  at  the  points  Pi,  Pi,  P3,  .  .  • 
Then  the  coordinates  of  Pi  are  {k,  h) ;  of  P2  are  {—  k,h);oi  P3  are 
(—  k,  —h  ),  etc.  Hence  sin  B  =  kja,  cos  B  =  hja,  sin  (90°  -\-  B)  = 
h  la,  cos  (90°  +  B)  =  -kja,  sin  (270°  -B)  =  -h/a,  cos  (270°  -  B) 
—  —  k/a,  etc.,  which  lead  to  the  equalities: 

sin  (90°  +  B)  =  cos  B  (1) 

cos  (90°  -{-  B)  =  -  sinB  (2) 

sin  (270°  -  B)  =  -  cosB  (3) 

cos  (270°  -  0)  -  -  sin  0  (4) 

etc. 


§162] 


TRIGONOMETRIC  EQUATIONS 


291 


By  division  of  (1)  by  (2)  and  (3)  by  (4), 

tan  (90°  4-  ^)  =  -  cot  ^  (5) 

tan  (270°  -  6)  =  cot  6  (6) 

Also  from  Fig.  a  16  A,  cos  (180°  -  d)  =  -  h/a,  sin  (180°  +  6) 

=  -  kfa,  cos  (180°  +  8)  =  -  h/a,  sin  (-  0)  =  -  k/a,  cos  (-  6) 

=  h/a, whence  there  results: 


sin  (180°  -  6)  =  sin  d 

(7) 

cos  (180°  -  ^)  =  -  cos  ^ 

(8) 

and  by  division 

tan  (180°  -  8)  =  -  tan  ^ 

(9) 

also 

sin  (180°  +  8)  =  -  sin  8 

(10) 

cos  (180°  +  (?)  =  -  cos  (9 

(11) 

and  by  division 

tan  (180°  +  8)  =  tan  8  (12) 

In  the  above  work  the  angle  8  is  drawn  as  an  angle  of  the  first 

quadrant.     The  proof  that  the  results  hold  for  all  values  of  8  is 

best  given  by  means  of  the  addition  formulas  of  §§160  and  161. 

The  method  will  be  outlined  in  the  next  section. 

The  results  are  brought  together  in  the  following  table.  No 
effort  should  be  made  to  commit  these  results  to  memory  in  this 
form.  The  statements  in  the  form  of  theorems  given  below  offer  a 
ready  means  of  remembering  all  of  the  results. 

TABLE  VII 

Functions   of  0   Coupled  with   an  Even  or  with  an  Odd   Number   of 

Right  Angles 


1     -0 

90°- e 

90°  +  9 

180°  -  9 

180°  +  5  270°-  9 

270°+  9 

sin 
cos 
tan 

—  sin  6 
cos  9 

-  tane 

cos  9          cos  9 
sin  9     —  sin  ^ 
cot  9      —  cot  0 

sin  9 

—  cos  9 

-  tan  9 

—  sin  9 

—  cos  9 
tane 

—  cos  9 

—  sin  9 
cot  9 

—  cos  5 
sin  9 

—  cot  9 

For  completeness  of  the  table  the  functions  of  ( —  8)  and  of 
(90°  —  8)  have  been  inserted  in  columns  1  and  2. 

All  of  the  above  results  can  be  included  in  two  simple  state- 
ments.    For  this  purpose  it  is  convenient  to  separate  into  different 


292        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§163 

classes  the  composite  angles  that  are  made  by  coupling  6  with 
an  odd  number  of  right  angles,  as  (90°  +  d),  {d  -  90°),  (270°  -  6), 
(450°  +  6),  etc.,  and  those  composite  angles  that  are  made  by- 
coupling  d  with  an  even  number  of  right  angles,  as  (180°  +  6), 
(180°  -  6),  (360°  -  d),{-  6),  etc.  Note  that  0  is  an  even  num- 
ber, so  that  (  —  6)  or  (0°  —  6)  falls  into  this  class  of  composite 
angles.    We  can  then  make  the  following  statements: 

Theorems  on  Functions  of  Composite  Angles 

Think  of  the  original  angle  d  as  an  angle  of  the  first  quadrant: 

I.  Any  function  of  a  composite  angle  made  by  coupling  6  (by 
addition  or  subtraction)  with  an  even  number  of  right  angles,  is 
equal  to  the  same  function  of  the  original  angle  6,  with  an  algebraic 
sign  the  same  as  the  sign  of  the  function  of  the  composite  angle  in 
its  quadrant. 

II.  Any  function  of  a  composite  angle  made  by  coupling  0  (by 
addition  or  subtraction)  with  an  odd  number  of  right  angles,  is  equal 
to  the  co-function  of  the  original  angle  6,  with  an  algebraic  sign 
the  same  as  the  sign  of  the  function  of  the  composite  angle  in  its 
quadrant. 

For  example,  let  the  original  angle  be  6,  and  the  composite  angle 
be  (180° +  0).  Then  any  function  of  {180°  +  6),  say 
tan  (180°  +  ^),  is  equal  to  +  tan  6,  the  sign  +  being  the  sign  of  the 
tangent  in  the  quadrant  of  the  composite  angle  (180°  +  6)  or 
third  quadrant.  Likewise  cot  (270°  +  6)  must  equal  the  negative 
co-function  of  the  original  angle,  or  —  tan  6,  the  algebraic  sign 
being  the  sign  of  the  cotangent  in  the  quadrant  of  the  composite 
angle  (270°  +  6),  or' fourth  quadrant.  In  the  above  work  it  has 
been  assumed  that  the  angle  6  is  an  angle  of  the  first  quadrant. 
The  results  stated  in  italics  are  true,  however,  no  matter  in 
what  quadrant  8  may  actually  lie. 

163.  Functions  of  Composite  Angles.  General  Proof:  All 
of  the  results  given  by  Table  VII  or  by  theorems  I  and  II  above 
can  be  deduced  at  once  from  the  addition  formulas,  with  the 
especial  advantage  that  the  proof  holds  for  all  values  of  the  angle 
6.    Thus,  write 

sin  (a  +  jS)  =  sin  a  cos  jS  +  cos  a  sin  /3  (1) 

cos  (a  +  /3)  =  cos  a-cos  /3  —  sin  a  sin  jS  (2) 


1164] 


TRIGONOMETRIC  EQUATIONS 


293 


Put  a  =  180°,  and  j8  =  ±  0;  then  (1)  and  (2)  become,  re- 
spectively: 

sin  (180°  ±  6)  =  T  sin  e  (3) 

cos  (180°  +  ^)  =  -  cos  (9  (4) 

Also  in  (1)  and  (2)  put  a  =  90°,  and  jS  =  ±  B,  then  (1)  and  (2) 
become,  respectively: 

sin  (90°  ±  6)  =       cos  d  (5) 

cos  (90°  +  (9)  =  +  sin  0  (6) 

In  a  similar  manner  all  of  the  results  given  in  the  table  may  be 
proved  to  be  true. 

164.  Angle  that  a  Given  Line  Makes  with  Another  Line.     The 
slope  m  of  the  straight    line  y  =  mx  -\-  b  is  the  tangent  of  the 


J 

rv^ 

y 

X 

o 

V 

\^ 

Li 

^1 

\ 

\u 

Fig.   117. — The  Angle  4>  that  a  Line  Li  makes  with  L2. 


direction  angle,  that  is,the  tangent  of  the  angle  that  the  line  makes 
with  OX.  If  Li  and  Li  are  any  two  lines  in  the  plane,  the  angle 
that  Li  makes  with  L2  is  the  positive  angle  through  which  L2 
must  he  rotated  about  their  point  of  intersection  in  order  that  L2 
may  coincide  with  Li.  Represent  the  direction  angles  of  two 
straight  lines 

y  =  mix  +  bi  (1) 

y  =  mzx  +  b2  (2) 

by  the  symbols  Oi  and  62.  Then,  through  the  intersection  of  the 
lines  pass  a  line  parallel  to  the  OX-axis,  as  shown  in  Fig.  117. 
Call  (j>  the  angle  that  the  line  Li  makes  with  L2;  that  is,  the  positive 
angle  through  which  L2,  considered  as  the  initial  line,  must  be 
turned  to  coincide  with  the  terminal  position  given  by  Li.     If 


294        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§164 

di  >  $2,  then  <f>  =  $1-  02,  but  if  $2  >  6^,  then  0  =  180°  -(^2  - 
^1).     In  either  case  (by  equations  (9),  §162,  and  (3),  §58): 


That  is: 


or, 


tan  0  =  tan  (^1  -  62) 

(3) 

^^^  ^       tan  Bi  -  tan  $2 

tan  <f>  =  ..—r-T ir-~. :r 

1  +  tan  di  tan  62 

(4) 

4.^^  J.        nil  —  m2 

IK\ 

The  condition  that  the  given  lines  (1)  and  (2)  are  parallel  is 
obviously  that 

mi  =  m2  (6) 

Thus  the  lines  y  =  hx  -\- 1  and  2/  =  5a;  -  11  are  parallel. 

The  condition  that  the  given  lines  (1)  and  (2)  are  perpendicular 
to  each  other  is  that  tan  0  shall  become  infinite;  that  is,  that  the 
denominator  of  (5)  shall  vanish.  Hence  the  condition  of  perpen- 
dicularity is 

1  +  mim2  =  0 
or, 

mi  =  -  ^  (7) 

m2 

Therefore,  in  order  that  two  lines  may  he  perpendicular  to  each 
other,  the  slope  of  one  line  must  be  the  negative  reciprocal  of  the  slope 
of  the  other  line. 

Thus  the  lines  y  =  (2/3)x  -  4  and  t/  =  -  (3/2)x  +  2  are 
perpendicular. 

Exercises 

1.  Find  the  tangent  of  the  angle  that  the  first  line  makes  with 
the  second  line  of  each  set : 

(a)  y  =  2x  +  d,  y  =  x  +  2. 

(b)  y  =  3x-  3,  y  =  2x  +  1. 

(c)  2/  =  4a;  +  5,  y  =  3x  —  4. 

(d)  y  =  lOx  -\-  1.  y  =  Ux  -  1, 

2.  Find  the  angle  that  the  first  line  of  each  pair  makes  with  the 
second: 


§165]  TRIGONOMETRIC  EQUATIONS  295 

{a)  y  =  X  +  5,  y  =  —  X  +  5. 

(6)  y  =  (1/2)0:  +  6,  y  =  -  2x. 

(c)  y  =  2x  +  4:,  y  =  X  +  1. 

(d)  2x-\-Sy  =  1,  (2/3)x  +  2/  =  1. 

(e)  2x  +  4?/  =  3,  Sx  +  Qy  =  7. 
(/)  2x  +  4y  =  3,  6x  -  3^/  =  7. 

3.  Find  the  angle,  in  each  of  the  following  cases,  that  the  first  line 
makes  with  the  second: 

(a)  y  =  x/\/3  +4,  y  =  VS  x  +  2. 

(b)  y  =  x/\/S  +  1,  2/  =  V3  a;  -  4. 

(c)  2/  =  V3  X  -  6,  2/  =  V3  a;  -  3. 

4.  Find  the  angle  that  2y  —  Qx  +  7  =0  makes  with   y  -\-  2x  -{- 
7  =  0  and  also  the  angle  that  the  second  line  makes  with  the  first. 

165.  The   Functions    of   the    Double   Angle.    The    addition 
formulas  for  the  sine,  cosine  and  tangent  reduce  to  formulas  of 
great  importance  for  the  special  case  ^  =  a. 
Thus:  sin  {a  -\-  a)  =  sin  a  cos  a  +  cos  a  sin  a 

or:  sin  2a  =  2  sin  a  cos  a  (1) 

Also:  cos  {a  -\-  a)  =  cos  a  cos  a  —  sin  a  sin  a 

which  can  be  written  in  the  three  forms : 

cos  2  a  =  cos2  a  —  sin^  a  (2) 

cos  2  o:  =  2  cos2  a  -  1  (3) 

cos  2  a  =  1  —  2  sin2  a  (4) 

Forms  (3)   and    (4)    are  obtained    from    (2)    by    substituting, 
respectively,  sin^  a  =  \  —  cos^  a  and  cos^  a  =  1  —  sin^  a. 
Equations  (3)  and  (4)  are  frequently  useful  in  the  forms: 

„  1  +  cos  2  a  ,-v 

cos^  a  = (5) 

2 

.  „           1  -  cos  2a  ,^x 

sin*  a  =  s (6) 


Again: 


or: 


tan  a  +  tan  a 

^  ^  ~  \  —  tan  a  tan  a 


2  tan  a  ,_, 

tan  2q:  =  , *— ?—  (7) 

1  -  tan^  a 


296        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§166 

166.  The  Functions  of  the  Half  Angle.     From  (6)  and  (5)  of 

§165  we  obtain,  after  replacing  a  by  ul2  and  extracting  the 

square  root, 

sm  (u/2)  =  ±  V(l  -  cosu)/2'  (1) 

cos  (u/2)  =   +  \/(r+  cos  u)  /2  (2) 

Dividing  (1)  by  (2),  we  obtain: 

\  1  +  cos  u  sin  u  -  1  +  cos  u       ^^ 

Formulas  (1),  (2)  and  (3)  have  many  important  applications  in 
mathematics.     As  a  simple  example,  note  that  the  functions  of  15° 
may  be  computed  when  the  functions  of  30°  are  known.     Thus: 
cos  30°  =  (1  /2)  V3 


therefore:      sin  15°  =  \/(l  -  cos  30°) /2  =  Vl/2  -  (1/4)  V3 
Also:  cos  15°  =  ^1/2  +  (1/4)  \7f 

Likewise  by  (5) : 

Exercises 

1.  Compute  sin  60°  from  the  sine  and  cosine  of  30°. 

2.  Compute  sine,  cosine,  and  tangent  of  22 1°. 

3.  If  sin  X  =  2/5,  find  the  numerical  value  of  sin  2x,  and  cos  2x 
tan  2x,  if  x  be  the  first  quadrant. 

4.  Show  by   expanding    sin   {x  +  2x)    that    sin  3a;  =  3  sin  a;  — 
4  sin^x. 

.    ^         X      o        3  tan  X  —  tan'  x 

5.  Prove  tan  6X  =  — ^ — —^z- — . 

1—3  tan2  X 

6.  Show  that  sin  2^/sin  d  —  cos  20/cos  0  =  2  sec  6. 

7.  Show  that: 


(  .    d  e\i 

Ism 2  +  C0S2)    =  1  +  sm  e. 


8.  Show  that:  cos  20(1  +  tan  26  tan  6)  =  1. 

9.  If  sin  A  =  3/5,  calculate  sin  (A/2). 

10.  Prove  that  tan  (7r/4  +  d)  =  ~^^^- 

1  —  tan  u 

11.  Prove  that  tan  (7r/4  -  0)  =  (1  -  tan  0)/(l  +  tan  6). 


§167]  TRIGONOMETRIC  EQUATIONS  297 

1  +  sin  0 


12.  Show  that  sec  0  +  tan  0  = 

13.  Show  that 


cos  d 
1  +  2  sin  a  cos  a       cos  a  +  sin  a 


cos^  a  —  sin^  a         cos  a  —  sin  a 

14.  Show  that  sec  6  +  tan  6  =  tan   7  +  o    * 

15.  Show  that  — 7— -i — j ,— ^  =  tan  A  tan  B. 

cot  A  +  cot  B 

16.  Prove  that  cos  (s  +  t)  cos  (s  —  i)  +  sin  (s  +  0  sin  (s  —  t)  = 
cos  2^ 

167.  Siuns  and  Differences  of  Sines  and  of  Cosines  Expressed 
as  Products.  The  following  formulas,  which  permit  the  substi-. 
station  of  a  product  for  a  sum  of  two  sines  or  of  two  cosines,  are 
important  in  many  transformations  in  mathematics,  especially  in 
the  calculus.  They  are  immediately  derivable  from  the  addition 
formulas;  thus,  by  the  addition  formulas  (14)  and  (16),  §160,  we 
obtain: 

sin  (a  +  6)  +  sin  (a  —  6)  =2  sin  a  cos  b 
Likewise  by  subtraction  of  the  same  formulas: 

sin  (a  +  6)  —  sin  (a  —  b)  =2  cos  a  sin  b 
By  the  addition  and  subtraction,  respectively,  of  the  addition 
formulas  for  the  cosine  there  results: 

cos  (a  +  6)  +  cos  (a  —  b)  =2  cos  a  cos  b. 

cos  (a  +  6)  —  cos  {a  —  b)  =  —  2  sin  a  sin  b. 
Represent  (a  +  6)  by  a  and  {a  —  b)  by  jS. 
Then  a  =  (a  +  18)  /2  and  b  =  (a  -  13) /2 
Hence  the  above  formulas  become: 

sin  a  +  sin  i3  =  2  sin  — ^— ^  cos  — ^  (1) 

2 

sin  a  —  sin  |8  =  2  cos  — - —  sin  — ^-^  (2) 

cos  a  +  cos  )8  =  2  cos  — ^-^  cos  — =-^  (3) 

cos  a  —  cos  jS  =  —  2  sin  — ^-^  sin  — ^-^  (4) 

The  principal  use  of  these  formulas  is  in  certain  transformations 
in  the  calculus.  A  minor  use  is  in  adapting  certain  formulas  to 
logarithmic  work  by  replacing  sums  and  differences  by  products. 


f 
298        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§169 

168.  Graph  of  y  =  sin  2x,  y  =  sin  nx,  etc.  Since  the  substi- 
tution of  nx  for  X  in  any  equation  multiplies  the  abscissas  of  the 
curve  by  1  jn,  or  {n  >  1)  shortens  or  contracts  the  abscissas  of  all 
points  of  the  curve  in  the  uniform  ratio  n:l,  the  curve  y  =  sin  2x 
must  have  twice  as  many  crests,  nodes  or  troughs  in  a  given 
interval  of  x  as  the  sinusoid  y  =  sin  x.  The  curve  y  =  sin  2x  is 
therefore  readily  drawn  from  Fig.  59  as  follows:  Divide  the  axis 
OX  into  twice  as  many  equal  intervals  as  shown  in  Fig.  59  and 
draw  vertical  lines  through  the  points  of  division.  Then  in  the 
new  diagram  there  are  twice  as  many  small  rectangles  as  in  the 
original .  Starting  at  0  and  sketching  the  diagonals  (curved  to 
fit  the  alignment  of  the  points)  of  successive  cornering  rectangles, 
the  curve  y  =  sin  2x  is  constructed.  It  is,  of  course,  the  ortho- 
graphic projection  of  y  =  sin  x  upon  a  plane  passing  through 
the  y-SLxis  and  making  an  angle  of  60°  (the  angle  whose 
cosine  is  1/2)  with  the  xy  plane.  The  curve  y  =  cos  2x  is  simi- 
larly constructed.  In  each  of  these  cases  we  see  that  the 
period  of  the  function  is  tt  and  not  27r. 

169.  Graph  of  p  =  sin  2^,  p  =  cos  26 j  etc.  The  curve  p  =  cos  ^ 
is  the  circle  of  diameter  unity  coinciding  in  direction  with  the  axis 
OX.  We  have  already  emphasized  that  as  6  varies  from  0°  to 
360°  the  circle  is  twice  drawn,  so  that  the  curve  consists  of  two 
superimposed  circular  loops.  Now  p  =  cos  26  will  be  found  to 
consist  of  four  loops,  somewhat  analogous  to  the  leaves  of  a  four- 
leafed  clover,  but  each  loop  is  described  but  once  as  6  varies  from 
0°  to  360°.  The  curve  p  =  cos  36  is  a  three-looped  curve,  but  each 
loop  is  twice  drawn  as  6  varies  from  0°  to  360°.  Also  p  =  cos  116 
has  eleven  loops,  each  twice  drawn,  while  p  =  cosl20  has 
twenty-four  loops,  each  one  described  but  once,  as  6  varies  from 
0°  to  360°. 

The  curves  p  =  cos  26,  p  =  sin  3^,  p  =  sin  ^  /2  should  be  drawn 
by  the  student  upon  polar  coordinate  paper. 

By  changing  the  scale  of  the  vectorial  angle,  the  circle  of  diame- 
ter unity  may  be  used  as  the  graph  of  the  equation  p  =  sin  n6. 
However  if  two  such  equations  are  to  be  represented  at  the  same 
time,  this  expedient  is  not  available,  for  the  vectorial  angles  of  the 
points  of  each  curve,  for  the  purpose  of  comparison,  must  be 
drawn  to  a  true  scale. 


(1701 


TRIGONOMETRIC  EQUATIONS 


299 


170.  Graph  of  y  =  sin^  x,  y  =  cos^  x.  The  graphs  y  =  sin^  x 
and  y  =  cos^  x  have  important  appUcations  in  science.  The  fol- 
lowing graphical  method  offers  an  easy  way  of  constructing  the 
curves  and  it  illustrates  a  number  of  important  properties  of  the 
functions  involved.  We  shall  first  construct  the  curve  y  =  cos^  x. 
At  the  left  of  a  sheet  of  8f  X  1 1-inch  paper,  draw  a  circle  of  radius 
36 
^    (=  2.30)  inches,  {OA,  Fig.  118).     Lay  off  the  angles  $  from 

OA,  Fig.  118,  as  initial  line,  corresponding  to  equal  intervals  (say 
10°  each)  of  the  quadrant  APE  as  shown  in  the  figure.  Let  the 
point  P  mark  any  one  of  these  equal  intervals.  Then  dropping 
the  perpendicular  AB  from  A  upon  OP,  the  distance  OB  is  the 


Y 
A 

S 

P>^^ 

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7^ 

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c  K. 

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/ 

\ 

nl    \\ 

\ 

/ 

\ 

/     ^^     1       \  ^^v. 

"1   ! 

/ 

\ 

/-"^A"V* 

i  1/ 

\ 

/ 

\ 

/ 

~— ~~  "^"^^^  \ 

4  J^ 

\ 

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"~^^-K_\ 

■^y] 

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y 

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Fig.  118. — The  Graph  oi  y  =  cos*  x. 

cosine  of  Q,  if  OA  be  called  unity.  Dropping  a  perpendicular  from 
B  upon  OA,  the  distance  OC  is  cut  off,  which  is  equal  to  05' 
or  cos*  B,  since  in  the  right  triangle  OB  A,  OB^  =  OCOA  =  OCl. 
Making  similar  constructions  for  various  values  of  the  angle  0, 
say  for  every  10°  interval  of  the  arc  APE,  the  line  OA  is  divided  at 
a  number  of  points  proportionally  to  cos^  9.  Draw  horizontal 
lines  through  each  point  of  division  of  OA.  Next  divide  the  axis 
OX  into  intervals  equal  to  the  intervals  of  6  laid  off  on  the  arc  APE; 
since  the  radius  of  the  circle  OA  was  taken  to  be  (36  /Stt)  inches,  an 
interval  of  10°  corresponds  to  an  arc  of  length  2/5  inch,  which 
therefore  must  be  the  length  of  the  equal  intervals  laid  off  on  OX. 
Through  each  of  the  points  of  division  of  OX  draw  vertical  lines, 


300        ELEMENTARY  MATHEMATICAL  ANALYSIS     [§171 

thus  dividing  the  plane  into  a  large  number  of  small  rectangles. 
Starting  at  A  and  sketching  the  diagonals  of  successive  cornering 
rectangles,  the  locus  ARS  oi  y  =  cos^  x  is  constructed. 

From  Fig.  118,  it  is  seen  that  B  always  lies  at  the  vertex  of  a 
right-angled  triangle  of  hypotenuse  OA.  Thus  asP  describes  the 
circle  of  radius  OA,  B  describes  a  circle  of  radius  OA  /2.  Therefore 
the  curve  ARSX  is  related  to  the  small  circle  ABO  in  the  same 
manner  that  the  curve  of  Fig.  59  is  related  to  its  circle;  conse- 
quently the  curve  ARSX  of  Fig.  118  is  a  sinusoid  tangent  to  the 
X-axis.  Thus  the  graph  y  =  cos^  x  is  a  cosine  curve  of  amplitude 
1  /2  and  wave  length  or  period  tt,  lying  above  the  x-axis  and  tangent 
to  it. 

In  Fig.  118,  OC  =  OH  -^  OC  =  OH  +  HB  cos  26  = 
1/2  -f-  (1/2)  cos  2(9.  Therefore  the  curve  ARS  has  also  the 
equation: 

2/ =  1/2-1- (1/2)  cos  2x  (1) 

Hence  we  have  a  geometrical  proof  that 

cos^x  =  1/2-f  (1/2)  cos2x  (2) 

which  is  formula  (5)  of  §165.  Note  that  (1)  is  the  curve  y  = 
cos  2x  with  its  ordinates  multiplied  by  1/2  then  translated  1/2 
unit  upward. 

The  curve  y  =  sin^  x  is  readily  drawn  in  a  manner  similar  to  that 
above,  by  laying  off  the  angle  6  from  OX  as  initial  line.     The  curve 

TT 

is  the  same  as  that  of  Fig.  118,  moved  the  distance  j  to  the  right. 

B.  PLANE  TRIANGLES :  CONDITIONAL  EQUATIONS 

171.  Law  of  Sines.  The  first  of  the  conditional  equations  per- 
taining to  the  oblique  triangle  is  a  proportion  connecting  the  sines 
of  the  three  angles  of  the  triangle  with  the  lengths  of  the  respect- 
ive sides  lying  opposite.  Call  the  angles  of  the  triangle  A,  B,  C, 
and  indicate  the  opposite  sides  by  the  small  letters  a,  b,  c,  respect- 
ively. From  the  vertex  of  any  angle,  drop  a  perpendicular  p 
upon  the  opposite  side,  meeting  the  latter  (produced  if  necessary) 
at  D.  Then,  from  the  properties  of  right  triangles,  we  have,  from 
either  Fig.  119  (1)  or  119  (2) 

p  =  csin  DAB  =  a  sin  (7  (1) 


§172] 


TRIGONOMETRIC  EQUATIONS 


301 


But, 

sin  DAB  =  sin  A  Fig.  119  (1) 

=  sin  (180°  -  A)  Fig.  119  (2) 
=  sin  A 
Therefore:  p  =  csinA  =  asinC  (2) 

Or:  a/sin  A  =  c/sinC  (3) 

In  like  manner,  by  dropping  a  perpendicular  from  A  upon  a,  we 
can  prove: 

b/sinB  =  c/sin  C  (4) 

Therefore:        a /sin  A  =  b/sin  B  =  c/sin  C  =  2R  (5) 

Stated  in  words,  the  formula  says:  In  any  oblique  triangle  the 
sides  are  proportional  to  the  sines  of  the  opposite  angles. 


Fig.   119. — Derivation  of  the  Law  of  Sines  and  the  Law  of  Cosines. 


Geometrically:  Calling  each  of  the  ratios  in  (5)  2R,  it  is  seen 
from  Fig.  119  (2)  that  R  is  the  radius  of  the  circumscribed  circle, 
and  that  c/sin  C  =  2R  can  be  deduced  from  the  triangle  BAE, 
Similar  construction  can  be  made  for  the  angles  B  and  A. 

172.  Law  of  Cosines.  From  plane  geometry  we  have  the  theo- 
rem: The  square  of  any  side  opposite  an  acute  angle  of  an  oblique 
triangle  is  equal  to  the  sum  of  the  squares  of  the  other  two  sides  di- 
minished by  twice  the  product  of  one  of  those  sides  by  the  projection 
of  the  other  side  on  it.     Thus  in  Fig.  119  (1) : 

a2  =  62  +  c2  -  2bd  (1) 

Now:  d  =  c  cos  A 

Therefore:  a^  =  b^  +  c"^  -  2bc  cos  A  (2) 


302        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§172 

Likewise  we  learn  from  geometry  that  the  square  of  any  side  oppo- 
site an  obtuse  angle  of  an  oblique  triangle  is  equal  to  the  sum  of 
the  squares  of  the  other  two  sides  increased  by  twice  the  product  of  one 
of  those  sides  by  the  projection  of  the  other  on  it.  Thus  in  Fig.  119 
(2): 

a2  =  62  +  c2  +  2bd  (3) 

Now:        d  =  c  cos  DAB  =  c  cos  (180  —  A)  =  —  c  cos  A 
Therefore  (3)  becomes: 

a2  =  b2  +  c2  -  2bc  cos  A  (4) 

This  is  the  same  as  (2),  so  that  the  trigonometric  form  of  the  geo- 
metricai  theorem  is  the  same  whether  the  side  first  named  is  oppo- 
site an  acute  or  opposite  an  obtuse  angle. 
In  the  same  way  we  may  show  that,  in  any  triangle: 

b2  =  c2  +  a2  -  2ca  cos  B  (5) 

c2  =  a2  +  b2  -  2ab  cos  C  (6) 

Independently  of  the  theorem  from  plane  geometry,  we  note  from 
Fig.  119(1): 

a^  =  (b  -  dY  -\-p^  =  {b  -  dy  +  c2  -  d2 

=  62  _(_  c2  -  2bd 

=  62  +  c2  -  26c  cos  A 
From  119  (2) :       a2  =  (6  +  d)2  +  p2  =  (6  +  (i)2  +  ^2  _  d^ 

=  62  +  c2  +  2bd 

=  62  -f  c2  +  26c  cos  DAB 

=  62  +  c2  -  26c  cos  A 
since  DAB  =  180°  -  A  and  cos  (180°  -  A)  =  -  cos  A 

Second  Proof:  Since  any  side  of  an  oblique  triangle  is 
the  sum  of  the  projections  of  the  other  two  sides  upon  it,  the 
angles  of  projection  being  the  angles  of  the  triangle,  we  have: 

a  =  b  cos  C  +  c  cos  B 

b  =  c  cos  A  +  a  cos  C  (7) 

c  =  a  cos  B  +  b  cos  A 

Multiply  the  first  of  these  equations  by  a,  the  second  by  6, 
the  third  by  c,  and  subtract  the  second  and  third  from  the  first. 
The  result  is; 


§173]  TRIGONOMETRIC  EQUATIONS  303 

^2  _  ^2  _  ^2  =  0^5  cQg  c  +  ca  cos  B 

—  be  cos  A  —  ab  cos  C 

—  ca  cos  B  —  be  cos  ^ 
=  —  26c  cos  A 

or:        a2  =  ^2  _^  c2  _  26c  cos  A 

173.  Law  of  Tangents.  An  important  relation  results  if  we 
take  formula  (5)  §171  by  composition  and  division.  First 
write  the  law  of  sines  in  the  form: 

a  _  sin  A 

6  ~  siiTB  ^^^ 

Then,  by  composition  and  division,  the  sum  of  the  first  anteced- 
ent and  consequent  is  to  their  difference  as  the  sum  of  the  second 
antecedent  and  consequent  is  to  their  difference;  that  is: 
a  4-  6  _  sin  A  +  sin  B 

a  —  b  ~  sin  A  —  sin  B  ^  ^ 

Expressing  the  sums  and  difference  on  the  right  side  of  (2)  by 
products  by  means  of  the  formulas  (1)  and  (2)  of  §167,  we 
obtain: 

a_±Jb  _  2  sin  ^{A  +  B)  cos  K^  -  B) 

a  -  6  ~  2  cos  ^(A  +  B)  sin  ^{A  -  B)  ^^^ 

or  simplifying  and  replacing  the  ratio  of  sine  to  cosine  by  the 
tangent,  we  obtain: 

a  +  b      tan  KA  +  B) 


(5) 


(6) 

Expressed  in  words:  In  any  triangle,  the  sum  oj  two  sides  is  to 
their  differenee,  as  the  tangent  of  half  the  sum  of  the  angles  opposite 
is  to  the  tangent  of  half  of  their  difference. 

Geometrical  Proof:  From  any  vertex  of  the  triangle  as 
center,  say  C,  draw  a  circle  of  radius  equal  to  the  shortest  of  the 
two  sides  of  the  triangle  meeting  at  C,  as  in  Fig.  120.  Let 
the  circle  meet  the  side  a  at  R  and  the  same  side  produced  at 


a-b 

tan 

KA- 

-B) 

In  like 

manner  it  follows  that: 

b  +  c 

tan 

KB  +  C) 

b  -c 

"  tan  KB  - 

-C) 

c-ha 

tan 

KC  +  A) 

c  —  a 

tan 

KC- 

-A) 

304        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§174 


E.    DTa.wAE,  AR,    Call  the  angles  at  ^,  a,  ^,  as  shown.    Then 
BE  =  a  -\-  b  and  BR  =  a  -  b.    Also: 

a-\-  P  =  A 
and:     Z  CRA  =  ^  -\-  B     (the     external    angle    of    a    triangle 
RAB  is  equal  to  the  sum  of  the  two  interior  opposite  angles), 
or  a  -  ^  =  B. 
Therefore: 


Draw  RS  \\  to  EA. 
By  similar  triangles: 


«  =  i(A  +  B) 
/3  =  KA  -  5) 
Z  EAR  =   Z  Ai^^S  =  90° 

BE/BR  =  AE/SR 
_AE^  SR 
~AR  '    AR 

Bui  BE  =  a  +  b  and  BR  =  a-b, 

while 

AE      ^  .SR'  ^ 

-j-n  =  tan  a  and  j^  =  tan  p 


Therefore; 


Fig.  120. — Geometrical  Deriva 
tion  of  Law  of  Tangents. 


a  +  b  _  tan   ^{A  +  B) 
a  —  b  ~  tan   ^(A  —  B) 
174.  The  following  special  form- 
ulas are  readily  deduced   from  the 
sine  formulas  and    are  sometimes 
useful  as  check  formulas  in  computation.     They  are   closely  re- 
lated to  the  law  of  tangents.     From  the  proportion: 

a:b:c  =  sin  A : sin  B : sin  C 
by  composition: 

c  sin  C 


a  -\-b      sin  A  +  sin  5 
Now  by  §165  (1)  and  §167  (1)  this  may  be  written: 
c  2  sin  |C  cos  ^C 


a  +  6       2  sin   |(A  +  B)   cos   ^{A  -  B) 
Since  C  =  180°  ~  {A  +  B),  therefore: 

(7/2  =  90°  -  h{A  +  B),  and  cos  C/2  =  sin  ^{A  +  B) 


Hence: 


c 
a  +  b 


sin  ^C 


cos  §(A  -  B) 


cos  ^A  +  B) 
cos  I  (A  -  B) 


(1) 


§175]  TRIGONOMETRIC  EQUATIONS  305 

In  like  manner  it  can  be  proved  that: 

__c sinMA  +  B) 

a  -  b  ~  sin  KA  -  B)  ^"^^ 

Both  (1)  and  (2)  can  be  readily  deduced  geometrically  from 
Fig.  120. 

175.  The  s-formulas.     The  cosine  formula: 
a2  =  62  _|_  c2  _  26c  cos  A 
can  be  written  in  the  forms: 

a2  =  (6  +  c)2  -  26c(l  +  cos  A)  (1) 

a2  =  (6  -  c)2  +  26c(l  -  cos  A)  (2) 

by  adding  (+  26c)  and  (—  26c)  to  the  right  member  in  each  case 

Now  we  know  from  §166,  (1)  and  (3),  that: 

1  +  cos  A  =  2  cos2  (A  12) 
1  -  cos  A  =  2  sin2  {A  /2) 

Therefore  (1)  and  (2)  above  become: 

a2  =  (6  +  c)2  -  46c  cos2  {A  /2)  (3) 

a2  =  (6  _  c)2  +  46c  sin2  (A  /2)  (4) 

writing  these  in  the  form: 

46c  sin2  (A  /2)  =  ci2  _  (6  -  c)^  (5) 

46c  cos2  {A  12)  =  (6  +  c)2  -  a2  (6) 

and  dividing  the  members  of  (5)  by  the  members  of  (6),  we 
obtain: 

tanM^/2)=|^;^-^:  (7) 

Factoring  the  numerator  and  denominator  we  obtain: 

tan^  {A  12)  =  («-±A^^K«-^l±4  (8) 

^    '  ^       (6  -f  c  +  a)  (6  +  c  —  a) 

Let  the  perimeter  of  the  triangle  be  represented  by  2s,  that  is, 

let: 

a  4-  6  +  c  =  2s 

Hence  subtracting  2c,  26,  and  2a  in  turn: 

a  +  6  —  c  =  2s  —  2c  (subtracting  2c) 
o  —  6  +  c  =  2s  —  26  (subtracting  26) 
6  +  c  —  a  =  2s  —  2a  (subtracting  2a) 

Therefore  equation  (8)  becomes: 

U.HAI2)  =  ^^^  (9) 

20 


306        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§175 
Let: 


then: 

or: 

Likewise: 


(s  —  a){s  —  h)  {s  —  c)  /s  = 

tan2  (A  12)  =  r^l{s  -  ay 

tan  (A/2)  =  r/(s  -  a) 

tan  (B/2)  =  r/(s  -  b) 
tan(C/2)  =r/(s  -  c) 


(10) 


(11) 

(12) 
(13) 


Fig.   121. — Geometrical  Derivation  of  the  s-Formulas. 

Geometrically:  These  formulas  may  be  found  by  means 
of  the  diagram  Fig.  121.  Let  the  circle  0  be  inscribed  in  the 
triangle  ABC;  its  center  is  located  at  the  intersection  of  the  bi- 
sectors of  the  internal  angles  of  the  triangle.     Let  its  radius  be  r. 


§177]  TRIGONOMETRIC  EQUATIONS  307 

Since  ATi  =  AT3,  BT2  =  BT3,  CTi  =  CT2,  and  since  2s  = 
a  +  6  +  c,  it  follows  that  one  way  of  writing  the  value  of  s  is: 

s  =  BTi  +  T2C  +  ATi 
Therefore: 

ATi  =  s  -  a 
Heitce  it  follows  that: 

tan(A/2)  =rl(s  -  a)'  (14) 

Since  this  result  is  the  same  as  (11)  above,  it  proves  that  the 
r  of  equation  (10)  is  the  radius  of  the  inscribed  circle,  and  there- 
fore proves  that  the  radius  of  the  inscribed  circle  may  be  expressed 
by  the  formula 

1(8-^(8 -~bj^^^) 


=^- 


(15) 


a  fact  that  is  usually  proved  in  text  books  on  plane  geometry. 

177.  *  Miscellaneous  Formulas  for  Oblique  Triangles.  The  fol- 
lowing formulas  are  given  without  proof.  They  are  occasionally 
useful  for  reference,  although  no  use  will  be  made  of  them  in 
this  book.  The  following  notation  is  used:  The  three  sides  of 
the  oblique  triangle  are  named  a,  b,  c,  and  the  angles  opposite 
these  A,  B,  C,  respectively.  The  semi-perimeter  of  the  triangle 
is  s,  or,  2s  =  a  -\-  b  -\-  c.  The  radius  of  the  circumscribed  circle 
is  R,  that  of  the  inscribed  circle  is  r,  and  the  radii  of  the  escribed 
circles  are  fo,  Vb,  Vc,  tangent,  respectively,  to  the  sides  a,  b,  c 
of  the  given  triangle.     K  stands  for  the  area  of  the  triangle. 

Then: 

s  =  4:R  cos  ^A  cos  iB  cos  ^C  (1) 

s  —  c  =  4J?  sin  ^A  sin  ^B  cos  ^C  (2) 

and  analogs  for  s  —  a  and  s  —  b. 

r  =  4:R  sin  ^A  sin  ^B  sin  ^C  (3) 

To  =  4/2  cos  |A  cos  ^B  sin  ^C  (4) 

and  analogs  for  ra  and  rt. 

Ta  =  s  tan  ^A,  n  =  s  tan  ^B,  Vc  =  s  tan  ^C  (5) 

2K  =  ab  sin  C  =  6c  sin  A  =  ca  sin  B  (6) 

K    =  2R^  sin  A  sin  B  sinC  =  ~~  (7) 


308        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§178 


K    =    Vs(s  -  a)(s  -  b){s  -  c)  (8) 

K    =  rs  =  ra{s  —  a)  =  rb{s  —  6)  =  rds  —  c)  (9) 

K2  =  rranvc  (10) 
K^  =  {s  -  a)  tan  ^A  =  {s  -  b)  tan  ^B  = 

(s-c)taniC  (11) 

C.  NUMERICAL  SOLUTION  OF  OBLIQUE  TRIANGLES 

178.  An  oblique  triangle  possesses  six  elements;  namely,  the 
three  sides  and  the  three  angles.  If  any  three  of  these  six 
magnitudes  be  given  (except  the  three  angles),  the  triangle  is 
determinate,  or  may  be  constructed  by  the  methods  explained 
in  plane  geometry;  it  will  also  be  found  that  if  any  three  of  these 
six  magnitudes  be  given,  the  other  three  may  be  computed  by  the 
formulas  of  trigonometry,  provided,  in  both  instances,  that  the 
given  parts  include  at  least  one  side. 

It  is  convenient  to  divide  the  solution  of  triangles  into  four 
cases,  as  follows: 

I.  Given  two  angles  and  one  side. 
II.  Given  two  sides  and  an  angle  opposite  one  of  them. 

III.  Given  two  sides  and  the  included  angle. 

IV.  Given  the  three  sides. 

The  solution  of  these  cases  with  appropriate  checks  will  now 
be  given.  The  best  arrangement  of  the  work  of  computation 
usually  consists  in  writing  the  data  and  computed  results  in  the 
left  margin  of  a  sheet  of  ruled  letter  paper  (8|  inches  X  11  inches) 
and  placing  the  computation  in  the  body  of  the  sheet.  Every 
entry  should  be  carefully  labeled  and  computed  results  should  be 
enclosed  in  square  brackets.  All  work  should  be  done  on  ruled 
paper  and  invariably  in  ink.  Special  calculation  sheets  (forms 
M2  and  M7)  have  been  prepared  for  the  use  of  students.  Neatness 
and  systematic  arrangement  of  the  work  and  proper  checking 
are  more  important  than  rapidity  of  calculation. 

179.  Computer's  Rules.  The  following  computer's  rules  are 
useful  to  remember  in  logarithmic  work: 

Last  Digit  Even:  When  it  becomes  necessary  to  discard  a 
5  that  terminates  any  decimal,  increase  by  unity  the  last  digit 


§180]  TRIGONOMETRIC  EQUATIONS  309 

retained  if  it  be  an  odd  digit,  but  leave  it  unchanged  if  it  be  an 
even  digit;  that  is,  keep  the  last  digit  retained  even.  Thus  log  x 
=  0.4971;  hence  write  (1/2)  log  tt  =  0.2486.  Also  log  sin 
18°  5'  =  9.4900  +  (correction)  19.5  =  9.4920. 

Of  course  if  the  discarded  figure  is  greater  than  5,  the  last 
digit  retained  is  increased  by  1,  while  if  the  discarded  figure  is 
less  than  5,  the  last  digit  retained  is  unchanged. 

Functions  of  Angles  in  Second  Quadrant:  In  finding 
from  the  table  any  function  of  an  angle  greater  than  100°  (but 
<  180°)  replace  by  their  sum  the  first  two  figures  of  the  number 
of  degrees  in  the  angle  and  take  the  cof unction  of  the  result.  The 
method  is  valid  because  it  is  equivalent  to  the  subtraction 
of  90°  from  the  angle.  By  §162  this  always  gives  the  cor- 
rect numerical  value  of  the  function.  The  algebraic  sign  should 
be  taken  into  account  separately.  Thus:  sin  157°  32'  1"  = 
cos  67°  32'  7".  In  case  of  an  angle  between  90°  and  100°, 
ignore  the  first  figure  and  proceed  in  the  same  way: 
tan  97°  57'  42"  =  -  cot  7°  57'  42" 

180.  Case  I.    Given  two  angles  and  one  side,  as  A,  B,  and  c. 

1.  To  find  C,  use  the  relation  A  +  5  +  C  =  180°. 

2.  To  find  a  and  6,  use  the  law  of  sines,  §171. 

3.  To  check  results,  apply  the  check  formula  (1)  or  (2)  §172. 

Example:  In  an  oblique  triangle,  let  c  =  1492,  A  =  49° 
52',  B  =  27°  15'.     It  is  required  to  compute  C,  a,  b. 

The  following  form  of  work  is  self  explanatory.  This  arrange- 
ment, while  readily  intelligible  to  the  beginner,  does  not  conform 
to  the  proper  standards  of  calculation  explained  above.  It 
should  be  noted,  however,  that  the  process  of  work  and  the  meaning 
of  each  number  entering  the  calculation  is  properly  indicated  or 
labeled  in  the  work. 

To  find  C:     C  =  180°  -  (A  +  5)  =  103°  53' 
To  find  a: 

As  sin  C  (103°  53')    colog  0.0129 

:  c    (1492)  log  3.1738 

::   sin  A  (49°  52')         log  9.8834 


a    [1175]  log  3.0701 


310        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§181 


To  find  h: 

As  sin  C  (103°  53')    colog  0.0129 
c    (1492)  log  3.1738 

sin  B  (27°  15')        log  9.6608 
b   [703.9]  log  2.8475 

Check:  As  sin   ^A  +  B)  (38°  33.5')      colog  0.2053 

:  sin  UA  -  B)  (11°  18.5')         log  9.2924 
: :  c  (1492)  log  3.1738 

:  a  -h  [469.4]  log  2.6715 

Also  a  —  h  from  first  computations  =  471.1  which  checks  469.4, 
as  computed,  within  1.7. 

The  above  work  arranged  in  compact  form  appears  as  follows: 

Computation  of  Triangle 
c,  A  and  B  given 


Data  and  results 

To  find  a 

To  find  b 

Check 

csin  A 
°-    sine 

c  sin  B 
'^  -    sin  ~C 

a  - 

c  sin  HA  -  B) 
sin  i{A  +  B) 

c  =  1492  log  3.1738 

A  =49°  52'  log  sin  9.8034 
B  =  27°  15' 

C  =  (103°  53')  colog  sin  0  .  0129 


log  3.1738  log  3. 1738 


log  sin  9 .  6608 
colog  sin  0.0129 


a  =  [1175]             log 

3.0701 

b  =  [703.9] 

log    2.8475 

(A  -  B)/2  =  (11°  19.5') 
{A  +B)/2  =  (38°  33.5') 

log  sin  9 .  2924 
colog  sin  0.2053 

a-b  =  (471.1) 

Check! 

[469.4]     2.6715 

Examples 

Find  the  remaining  parts,  given: 

1.  A  =  47°  20',  B  =  32°  10', 

2.  B  =  37°  38',  C  =  77°  23', 
Z.    B  =  25°  2',               C  =  105°  17', 
4.     C  =  19°  35',            A  =  79°  47', 
181.  Case  II.     Given  two  sides  and 

them,  as  a,  b,  and  A. 


a  =  739. 
b  =  1224. 
b  =  0.3272. 
c  =  56.47. 

an  angle  opposite   one  of 


1.  To  find  B,  use  the  law  of  sines,  §171. 

2.  To  find  C,  use  the  equation  A  -{-  B  -\-  C  =  180°. 

3.  To  check,  apply  the  check  formula  (1)  or  (2),  §174. 
When  an  angle  as  B,  above,  is  determined  from  its  sine,  it  admits 


§181] 


TRIGONOMETRIC  EQUATIONS 


311 


of  two  values,  which  are  supplemeijtary  to  each  other.  There 
may  be,  therefore,  two  solutions  to  a  triangle  in  Case  II.  The 
solutions  are  illustrated  in  Fig.  122. 

In  case  one  of  the  two  values  of  B  when  added  to  the  given 
angle  A  gives  a  sum  greater  than  two  right  angles,  this  value 
of  B  must  be  discarded,  and  but  one  solution  exists.  If  a  be 
less  than  the  perpendicular  distance  from  C  to  c,  no  solution 
is  possible. 

,c 


Fig.   122. — Case  II  of  Triangles,  for  One,  Two,  and  Impossible  Solutions- 

Example:    Find  all  parts  of  the  triangle  if  a  =  345,  b  =  534, 
and  A  =  25^  25'. 
The  solution  is  readily  understood  from  the  following  work. 


To  find  5: 

As     a              (345) 

colog    7.4622 

:     b             (534) 

log    2.7275 

: :  sin  A  (25°  25') 

log    9.6326 

:    sin  B  [41°  37'] 

log    9.8223 

5'  [138°  23'] 

C  =  180" 

-  (A  +  5)  =  112°  58'. 

To  find  c: 

As  sin  A  (25°  25') 

colog    0.3674 

:  sin  C  (112°  58') 

log    9.9641 

: :  a         (345). 

log    2.5378 

:  c         [740.1] 

log    2.8693 

Check: 

As  c                        (740.1) 

log  2.8693 

:  b  -a                 (189) 

colog  7.7235 

::  sin  ^(B  +  A)  (33°  31') 

colog  0.2579 

:  sin  hiB  -  A)  [8°  6'] 

log  9.1489 

Check! 

9.9996 

312        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§181 
The  sum  of  the  logs  should  be  0.     The  discrepancy  is  4  in  the  last 


decimal  place. 

To  find  c': 

C 

'  =  180°  -  A  - 

B'  =  16°  12' 

As  sin 

A 

(25°  250 

colog  0.3674 

:  sin 

C 

'  (16°  12') 

log  9.4456 

: :  a 

(345) 

log  2.5378 

:c' 

[224.3] 

log  2.3508 

To  Check: 

Asc' 

(224.3) 

log  2.3508 

:h  -a 

(189) 

colog  7.7235 

::  sini(5'  +  A) 

(81°  54') 

colog  0.0043 

:  sin|(B'  -A) 

(56°  29') 

log  9.9210 

Check!  9.9996 

The  following  arrangement  of  the  work  satisfies  the  require- 
ments of  properly  arranged  computation  and  is  much  to  be  pre- 
ferred to  the  arrangement  given  above. 

Computation  of  Triangle 
a,  b,  and  A  given 


Data  and 
results 

To  find  B 

To  find  c 

Check 

.     „     6  sin  A 
smB=      ^ 

a  sin  C 
^~  sin  A 

c        sin|(B  +  A) 
6-o~sin|(5- A) 

a  =  345  colog  7.4622  log  2.5378 

b  =  534  log  2.7275 

A  =  25°  25'  log  sin  9.6326  colog  sin  0.3674 


B  =  [41°  37']  9.8223 

C  =  (112°  58') log  sin  9.9641 

c  =  [740.1]  log  2.8693     log  2.8693 

{B  -  A)  12    =  (80°  6')  log  sin  9.1489 

{B  +  A)/2    =  (33°  31')  colog  sin  0.2579 

b  -  a  =  (189) colog  7.7235 

Check!  9.9996 

B'  =  (138°  23') 

C  =  (16°  12')  log  sin  9.4456 

c'  =  [224.3]  log  2.3508  log  2.3508 

{B'  -  A)  12   =  (56°  29')  log  sin  9.9210 

{B'  +  A)/2   =  (81°  54')  colog  sin  0.0043 

b  -  a  =  (.189)  colog  7.7235 

Check!  9.9996 


1. 

a 

=  0.8, 

2. 

a 

=  17.81, 

3. 

h 

=  81.05, 

4. 

c 

=  50.37, 

6. 

a 

=  1213, 

182.  1 

Case   III. 

il82]  TRIGONOMETRIC  EQUATIONS  313 

Examples 

Compute  the  unknown  parts  in  each  of  the  following  triangles: 

b  =  0.7,  B  =  40°  15'. 

b  =  11.87,  A  =  19°  9'. 

c  =  98.75,  C  =  99°  19'. 

a  =  58.11,  C  =  78°  13'. 

b  =  1156,  B  =  94°  15'. 

Given   two   sides   and   the   included   angle,    as 
a,  b,  C. 

1.  To  find  A  -h  B,  use  A  +  B  =  180°  -  C. 

2.  To  find  A  and  B,  compute  {A  —  B)  /2  by  the  law  of  tangents, 
§173,  equation  (4),  then  A  =  {A -\-  B)  12  +  {A  -  B)  /2  and 
B  =  {A^B)I2  -  {A  -  B)  12. 

3.  To  find  c,  use  law  of  sines,  §171. 

4.  To  check,  use  the  check  formula  (2)    §174. 
Example:     Given  a  =  1033,  b  =  635,  C  =  38°  36' 

A  +  B  =  180°  -  38°  36'  =  141°  24' 
To  find  A  andB: 

As  a  +  6  (1668)  colog     6.7778 

:  a  -b  (398)  log     2.5999 

: :  tan  hU  +  B)/       (70°  42')  log  tan     0.4557 


:  tan  K^  - 

B)/ 

[34°  16'] 
A  =  104° 
B  =36° 

58' 
26' 

log  tan 

9.8334 

To  find  c: 

As  sin  A 

(104°  58') 

colog 

0.0150 

:  sin  C 

(38°  36') 

log 

9.7951 

: :  a 

(1033) 

log 

3.0141 

:  c 

[667.1] 

log 

2.8242 

Check: 

As  sin  ^{A  -  B) 

(34°  16') 

colog 

0.2495 

:  sin  hU  +  B) 

(70°  42') 

log 

9.9749 

::a  -b 

(398) 

log 

2.5999 

:     c  [667.2]  log     2.8243 

Check! 

An  experienced  computer  would  arrange  the  above  work  as 
follows: 


314        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§183 

Computation  of  Triangle 
a,  h,  C  given 


Data  and  results 

To  find  A 

-B    To  find  C           Check 

t&n  HA- B) 
tanUA+B) 

a  —  b           o  sin  c           c           sinHA+B) 
a+b   '^        sin  A        a-b       sin^{A-B) 

a  =  1033  log  3.0141 

h  =  635 
a  -h  =  (398)  log  2.5999  log  2.5999 

a  +  6  =  (1668)  colog  6.7778 

C  =  38°  35'  log  sin  9.7951 

(A  +  B)/2    =  (70°  42')   log  tan  0.4557  log  sin  9.9749 


(A  -  B)/2    =  [34°  16']    log  tan  9.8334  colog  sin  0.2495 


A  =  (104°  58')  colog  sin  0.0150 

B  =  (36°  26') 

c  =  [667.1]  log  2.8242  log    2.8243 

Check! 

Examples 

Compute  the  unknown  parts  in  each  of  the  following  triangles. 

1.  a  =  78.9,  5  =  68.7,  C  =  78°  10'. 

2.  c  =  70.16,  a  =  39.14,  5  =  16°  16'. 

3.  6  =  1781,  c  =  982.7,  A  =  123°  16'. 

4.  a  =  TT  b  =  7r/2,  C  =  7r/3. 

183.  Case  IV.    Given  the  three  sides. 

1.  To  find  the  angles,  use  the  s-formulas,  §175,  (11),  (12) 
and  (13). 

2.  To    check,    use    A  -\- B  -\- C  =  180°. 

Example:  Given  a  =  455,  h  =  566,  c  =  677,  find  A,  B 
and  C. 

The  following  work  is  self  explanatory.  The  work  is  arranged 
in  final  compact  form,  which,  in  this  case,  is  as  simple  as  any- 
other  possible  arrangement. 


§183]  TRIGONOMETRIC  EQUATIONS  315 

Computation  of  Triangle 
a,  h,  c  given 

Data  and  results  To  find  A,  B,  C 


«2  =  (s  _  a){s  -  b){s  -  c)/i 
tan  A/2  =  r/(s  —a).  .  . 


a  =  455 

b  =  566 

c  =  677 

2s  =  1698 

s  =  849 

-  a  =  394 

-  6  =  283 

-  c  =  172 
r2 

r 

A/2  =  [20*^  53'] 

B/2  =  [27°  58': 

C/2  =  [41°  9'] 

A  =  41°  46' 

B  =  55°  56' 

C  =  82°  18' 


colog 

7.0711 

log 

2.5955 

log 

2.4518 

log 

2.2355 

log 

4.3539 

log 

2.1770 

log  tan 

9.5815 

log  tan 

9.7252 

log  tan 

9.9415 

Check!  180°  0' 


Exercises 

Find  the  values  of  the  angles  in  each  of  the  following  triangles: 

1.  a  =  173,  b  =  98.6,  c  =  230. 

2.  a  =  8.067,  b  =  1.765,  c  =  6.490. 
Z.     a  =  1911,                  b  =  1776,  c  =  1492. 

Miscellaneous  Problems 

The  instructor  will  select  only  a  limited  number  of  the  following 
problems  for  actual  computation  by  the  student.  The  student  should 
be  required,  however,  to  outline  in  writing  the  solution  of  a  number 
of  problems  which  he  is  not  required  actually  to  compute,  and,  when 
practicable,  to  block  out  a  suitable  check  for  each  one  of  them. 

1.  From  one  corner  P  of  a  triangular  field  PQR  the  side  PQ  bears 
N.  10°  E.  100  rods.  QR  bears  N.  63°  E.  and  PR  bears  N.  38°  10'  E. 
Find  the  perimeter  and  area  of  the  field. 


316        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§183 

2.  The  town  B  lies  15  miles  east  of  A,  C  lies  10  miles  south  of  A. 
X  lies  on  the  line  BC,  and  the  bearing  of  AX  is  S.  46°  20'  E.  Find 
the  distances  from  X  to  the  other  three  towns. 

3.  To  find  the  length  of  a  lake  (Fig.  123),  the  angle  C  =  48°  10', 
the  side  a  =  4382  feet,  and  the  angle  B  =  62°  20'  were  measured. 
Find  the  length  of  the  lake  c,  and  check. 

4.  To  continue  a  line  past  an  obstacle 
L,  Fig.  124,  the  line  BC  and  the  angles 
marked  at  B  and  C  were  measured  and 
found  to  be  1842  feet,  28°  15',  and  67° 
24',  respectively.  Find  the  distance  CD, 
and  the  angle  at  D  necessary  to  continue 
the  line  AB]  also  compute  the  distance 
BD. 

5.  Find  the  longer  diagonal  of  a  par- 
allelogram, two  sides  being  69.1  and  97.4 
and  the  acute  angle  being  29°  34'. 

What  is  the  magnitude  of  the  single 
force  equivalent  to  two  forces  of  69.1  and 
97.4  dynes  respectively,  making  an  angle  of  29°  34'  with  each  other? 

6.  A  force  of  75.2  dynes  acts  at  an  angle  of  35°  with  a  force  F. 
Their  resultant  is  125  dynes.     What  is  the  magnitude  of  i^? 

7.  The  equation  of  a  circle  is  p  =  10  cos  9.  The  points  A  and 
B  on  this  circle  have  vectorial  angles  31°  and  54°  respectively.  Find 
the  distance  AB,   (1)   along  the  chord;  (2)  along  the  arc  of  the  circle 


123. — Diagram  for 
Problem  3. 


Fig.   124. — Diagram  for  Problem  4. 


8.  Find  the  lengths  of  the  sides  of  the  triangle  enclosed  by  the 
straight  lines : 

e  =  26°;  d  =  115°;  p  cos  {d  -  45°)  =  50. 

9.  A  gravel  heap  has  a  rectangular  base  100  feet  long  and  30  feet 
wide.  The  sides  have  a  slope  of  2  in  5.  Find  the  number  of  cubic 
yards  of  gravel  in  the  heap. 


§183]  TRIGONOMETRIC  EQUATIONS  317 

10.  A  point  B  is  invisible  and  inaccessible  from  A  and  it  is  necessary 
to  find  its  distance  from  A.  To  do  this  a  straight  line  is  run  from  A 
to  P  and  continued  to  Q  such  that  B  is  visible  from  P  and  Q.  The 
following  measurements  are  then  taken:  AP  =  2367  feet;  PQ  =  2159 
feet;  APB  =  142°  37'.3;  AQB  =  76°  13'.8.     Find  AB. 

11.  To  determine  the  height  of  a  mountain  the  angle  of  elevation 
of  the  top  was  taken  at  two  stations  on  a  level  road  and  in  a  direct 
line  with  it,  the  one  5280  yards  nearer  the  mountain  than  the  other. 
The  angles  of  elevation  were  found  to  be  2°  45'  at  the  further  station 
and  3°  20'  at  the  nearer  station.  Find  the  horizontal  distance  ot  the 
mountain  top  from  the  nearer  station  and  the  height  of  the 
mountain  above  it.     Use  S  and  T  functions. 

12.  Explain  how  to  find  the  distance  between  two  mountain  peaks 
Ml  and  M^,  (1)  when  A  and  B  at  which  measurements  are  taken 
are  in  the  same  vertical  plane  with  Mi  and  M2;  (2)  when  neither  A 
nor  B  is  in  the  same  vertical  plane  with  Mi  and  Mi. 

13.  The  sides  of  a  triangular  field  are  534  yards,  679  yards  and 
474  yards.  The  first  bears  north,  and  following  the  sides  in  the 
order  here  given  the  field  is  always  to 
the  left.  Find  the  bearing  of  the  other 
two  sides  and  the  area. 

14.  From  a  triangular  field  whose  sides 
are  124  rods,  96  rods,  and  104  rods  a 
strip  containing  10  acres  is  sold.  The 
strip  is  of  uniform  width,  having  as  one 
of  its  parallel  sides  the  longest  side  of 
the  field.     Find  the  width  of  the  strip.  

15.  Three  circles  are  externally  mutu-  *    Problem  16^ 
ally  tangent.     Their  radii  are  5,  6,  7  feet. 

Find  the  area  and  perimeter  of  the  three-cornered  area  enclosed  by 
the  circles  and  the  length  of  a  wire  that  will  enclose  the  group  of 
three  circles  when  stretched  about  them. 

16.  To  find  the  distance  between  two  inaccessible  objects  C  and  Z), 
Fig.  125,  two  points  A  and  B  are  selected  from  which  both  objects  are 
visible.  The  distance  AB  is  found  to  be  7572  feet.  The  following 
angles  were  then  taken: 


ABD 

= 

122° 

37' 

ABC 

= 

70° 

12' 

BAG 

= 

80° 

20' 

BAD 

= 

27° 

13' 

Find  the  distance  DC  and  check. 


318        ELEMENTARY  MATHEMATICAL  ANALYSIS      l§183 


17.  A  circle  of  radius  a  has  its  center  at  the  point  (pi,  di).  Find 
its  equation  in  polar  coordinates.     (Use  law  of  cosines.) 

18.  A  surveyor  desired  the  distance  of  an  inaccessible  object  0 
from  A  and  B,  but  had  no  instruments  to  measure  angles.  He 
measured  AA'  in  the  line  AO,  BB'  in  the  line  BO]  also  AB,  BA',  AB'. 
How  did  he  find  OA  and  05? 

19.  From  a  point  A  a  distant  object  C  bears  N.  32°  16'  W.  with 
angle  of  elevation  8°  24';  from  B  the  same  object  bears  N.  50°  W. 
AB  bears  N.  10°  38'  W.  The  distance  AB  is  1000  yards.  Find  the 
distance  AC. 


Fig.   126. — Diagram  for  Problem  20. 

20.  The  angle  of  elevation  of  a  mountain  peak  is  observed  to  be 
19°  30'.  The  angle  of  depression  of  its  image  reflected  in  a  lake  1250 
feet  below  the  observer  is  found  to  be  34°  5'.  Find  the  height  of  the 
mountain  above  the  observer  and  the  horizontal  distance  to  it.  (See 
Fig.  126.) 

21.  One  side  of  a  mountain  is  a  smooth  eastern  slope  inclined  at  an 
angle  of  26°  10'  to  the  horizontal.  At  a  station  A  a  vertical  shaft  is 
sunk  to  a  depth  of  300  feet.  From  the  foot  of  the  shaft  two  horizontal 
tunnels  are  dug,  one  bearing  N.  22°  30'  E.  and  the  other  S.  65°  E. 
These  tunnels  emerge  at  B  and  at  C  respectively.  Find  the  lengths 
of  the  tunnels  and  the  lengths  of  the  sides  of  the  triangle  ABC. 


§183]  TRIGONOMETRIC  EQUATIONS  319 

22.  A  rectangular  field  ABCD  has  side  AB  =  40  rods;  AD  =  80 
rods.  Locate  a  point  P  in  the  diagonal  AC  so  that  the  perimeter  of 
the  triangle  APB  will  be  160  rods.  {Hint:  Express  perimeter  as  a 
function  of  angle  at  P.) 

23.  Find  the  area  enclosed  by  the  lines  y  =  o'  y  ^  v  ^  ^'  ^^^  ^^^ 
circle  x^  —  lOx  -{-  y^  =  0.     {Hint:  Change  to  polar  coordinates.) 

24.  The  displacement  of  a  particle  from  a  fixed  point  is  given  by 

d  =  2.5  cos  t  +  2.5  sin  t. 

What  values  of  t  give  maximum  and  minimum  displacements;  what 
is  the  maximum  displacement? 

26.  A  quarter  section  of  land  is  enclosed  by  a  fence.  A  farmer 
wishes  to  make  use  of  this  fence  and  60  rods  of  additional  fencing  in 
making  a  triangular  field  in  one  corner  of  the  original  tract.  Find  the 
field  of  greatest  possible  area.  Show  that  it  is  also  the  field  of  maxi- 
mum perimeter,  under  the  conditions  given. 

26.  A  force  Fi  =  100  dynes  makes  an  angle  of  6°  with  the  horizontal, 
and  a  second  force  F2  =  50  dynes  makes  an  angle  of  90°  with  Fi. 
Determine  6  so  that  (1)  the  sum  of  the  horizontal  components  of  Fi 
and  F2  shall  be  a  maximum;  (2)  so  that  the  sum  of  the  vertical  com- 
ponents shall  be  zero. 

27.  Find  the  area  of  the  largest  triangular  field  that  can  be  enclosed 
by  200  rods  of  fence,  if  one  side  is  70  rods  in  length. 

28.  Change  the  equation  of  the  curve  xy  =  I  io  polar  coordinates, 
rotate  through  —  45°  and  change  back  to  rectangular  coordinates. 

29.  A  particle  moves  along  a  straight  line  go  that  the  distance 
varies  directly  as  the  sum  sin  t  +  cos  t.  When  t  =  7r/4,  the  distance 
is  10;  find  the  equation  of  motion. 

30.  From  the  top  of  a  lighthouse  60  feet  high  the  angle  of  de- 
pression of  a  ship  at  anchor  was  observed  to  be  4°  52',  from  the 
bottom  of  the  lighthouse  the  angle  was  4°  2'.  Required  the  horizon- 
tal distance  from  the  lighthouse  to  the  ship  and  the  height  of  the 
base  of  the  lighthouse  above  the  sea. 

31.  A  vertical  square  shaft  measuring  3  feet  6  inches  on  a  side 
meets  a  horizontal  rectangular  tunnel  6  feet  6  inches  high  by  3  feet 
6  inches  wide.  Find  an  expression  for  the  length  of  a  line  AB 
shown  in  Fig.  127  when  the  angle  d  is  37°. 

32.  University  Hall  casts  a  shadow  324  feet  long  on  the  hillside 
on  which  it  stands.  The  slope  of  the  hillside  is  15  feet  in  100  feet, 
and  the  elevation  of  the  sun  is  23°  27'.  Find  the  height  of  the 
building. 


320 


ELEMENTARY  MATHEMATICAL  ANALYSIS 


§183 


33.  To  determine  the  distance  of  a  fort  A  from  a  place  B,  a  line  BC 
and  the  angles  ABC  and  BCA  were  measured  and  found  to  be  3225.5 
yards,  60°  34',  and  56°  10'  respectively.     Find  the  distance  AB. 

34.  A  balloon  is  directly  over  a  straight  level  road,  and  between 
two  points  on  the  road  from  which  it  is  observed.  The  points  are 
15,847  feet  apart,  and  the  angles  of  elevation  are  49°  12'  and  53°  29'. 
Find  the  height. 


Fig.   127.— Diagram  for  Problem  31. 


35.  Two  trees  are  on  opposite  sides  of  a  pond.  Denoting  the  trees 
by  A  and  B,  we  measure  AC  =  297.6  feet,  BC  =  864.4  feet,  and  the 
angle  ABC  =  87°  43'.     Find  AB. 

36.  Two  mountains  are  9  and  13  miles  respectively  from  a  town, 
and  they  include  at  the  town  an  angle  of  71°  36'.  Find  the  distance 
between  the  mountains. 

37.  The  sides  of  a  triangular  field  are,  in  clockwise  order,  534 
feet,  679  feet,  and  474  feet;  the  first  bears  north;  find  the  bearings 
of  the  other  sides  and  the  area. 

38.  Under  what  visual  angle  is  an  object  7  feet  long  seen  when 
the  eye  is  15  feet  from  one  end  and  18  feet  from  the  other? 

39.  The  shadow  of  a  cloud  at  noon  is  cast  on  a  spot  1600  feet 
west  of  an  observer,  and  the  cloud  bears  S.,  76°  W.,  elevation  23°. 
Find  its  height. 


CHAPTER  X 
WAVES 

184.  Sipiple  Harmonic  Motion.  Let  P  be  any  point  on  a  circle, 
and  let  D  be  the  projection  of  P  on  any  straight  line  in  the  plane 
of  the  circle.  Then  if  the  point  P  move  uniformly  (that  is,  so  that 
equal  distances  are  described  in  equal  times)  on  the  circle,  the 


\c£ 

B 

\ 

\m^ 

0  Hrk^^ 
\    w 

— *.                                   /^                                 IT 

1 

7hW--------"'~-'^'- 

1 1 1 1 1 1  N-fflifMiLI  1 1 1 1 

//3-^ 

KT                          IN. 

7                                S 

/  ^-'  T[      C 

"  —      Jl          S   •  ^2                                                   ^ 

EL      '^Z                                                             ^     N\ 

1 

■~  --     u  r  1     ^ 

z                                 \ 

y 

^\^3             S 

^^>^          s^           ^7 

vi                      rK                UrT 



H 

1                                  2. 

Fig.  128.- 


-Mechanical  Generation  of  Simple  Harmonic  Motion,  and  of  a 
Simple  Progressive  Wave. 


back-and-forth  motion  of  the  point  D  on  the  given  straight  line 
is  called  simple  harmonic  motion.  On  account  of  the  frequency 
with  which  this  term  will  occur,  we  shall  abbreviate  it  by  the 
symbols  S.H.M.  Fig.  128  illustrates  a  way  in  which  this 
motion  may  be  described  by  mechanical  means.  Let  the  uni- 
formly rotating  wheel  OAB  be  provided  with  a  pin  M  attached 
to  its  circumference,  and  free  to  move  in  the  slot  of  the  cross- 
head  as  shown,  the  arm  of  the  cross-head  being  restricted  to 
21  321 


322        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§184 

vertical  motion  by  suitable  guides  GiGi.  Then,  as  the  wheel 
rotates,  any  point  P  of  the  arm  of  the  cross-head  describes  simple 
harmonic  motion  in  a  vertical  direction.  The  amplitude  of 
the  S.H.M.  is  the  radius  of  the  circle,  or  OB;  its  period  is  the 
time  required  for  one  complete  revolution  of  the  wheel. 

In  elementary  physics  it  is  explained  that  the  motion  of  a  simple 
pendulum  is  nearly  simply  harmonic.  Also  that  the  motion  of 
a  point  of  a  vibrating  violin  string,  or  of  a  point  of  a  tuning  fork,  is 
S.H.M.  S.H.M.  is  a  fundamental  mode  of  motion  of  the  particles 
of  all  elastic  substances,  and  is  therefore  of  great  importance. 

The  motion  of  the  point  D2  can  readily  be  expressed  by  an  equa- 
tion, if  the  value  of  the  angle  B  be  expressed  in  terms  of  the  elapsed 
time  t.  Since  the  rotation  is  uniform,  B  =  kt,  where  k  is  the  angle 
described  in  one  second,  or  the  angular  velocity  of  P.  Let  the 
radius  OM  of  the  circle  be  a  feet.  If  0  be  taken  as  origin,  and  if 
the  angle  AOM  be  called  B,  then  if  the  point  M  was  at  A  when 
t  =  0,  the  displacement  OD2  =  ?/  is  given  at  any  time  t  by 

y  =  a  sin  ^  =  a  sin  kt  (1) 

In  a  similar  way,  the  point  Di,  the  projection  of  M  on  OA,  de- 
scribes a  S.H.M.,  and  the  displacement  ODi  =  x  may  be  written 
X  =  a  cos  B  =  a  cos  kt  (2) 

If  the  point  M  was  at  E  when  t  =  0,  the  displacements  OD2  =  y 
and  ODi  =  x  are  given  by 

y  =  a  sin  (kt  -  e)  (3) 

X  =  a  cos  (kt  —  e)  (4) 

when  e  stands  for  the  angle  EOA ;  for  kt  =  angle  EOM  and  B  = 

EOM  —  EOA.     In  this  equation  kt  —  e  h  called  the  phase  angle 

and  €  is  called  the  epoch  angle  of  the  S.H.M. 

These  expressions  may  also  be  written  in  terms  of  the  linear 
velocity  7  of  M  instead  of  the  angular  velocity  k  of  OM.  Let 
the  uniform  velocity  of  M  be  y  feet  per  second.  Since  the  radius 
OM  is  a  feet,  OM  rotates  at  the  rate  of  v/a  =  k  radians  per  second. 
This  value  of  k  may  be  substituted  in  equations  (1)  to  (4). 
It  is  obvious  that 

y  =  a  cos  kt 

represents    a     S.H.M.   ^    ^^    advance    of    y  =  a  sin  kt,   since 


§185]  WAVES  323 

sin  (kt  +  o)  =  ^^^  ^^'  ^  P^ir  ^^  S.H.M/s  possessing  this  prop- 
erty are  said  to  be  in  quadrature.  (1)  and  (2),  or  (3)  and  (4) 
may  be  said  to  be  in  quadrature. 

The  period  of  the  S.H.M.  y  =  a  sin  Jet  is  the  time  T  required 
for  a  complete  revolution.  If  ^i  be  the  time  at  which  M  is  at  any 
given  position,  and  if  ^2  be  the  time  at  which  M  is  next  at  the 
same  position,  then,  since  the  angular  velocity  multiplied  by 
the  elapsed  time  gives  the  angular  displacement,  we  have, 

k{t2  -  ti)  =  2ir 
Therefore,  since  the  difference  t2  —  ti  =  T  is  the  period: 

n,       27r 

The  number  of  complete  periods  per  unit  time  is: 

^^        T       27r 
N  is  called  the  frequency  of  the  S.H.M. 

It  is  obvious  that  all  points  of  the  moving  cross-head,  Fig. 
128,  describe  S.H.M.,  and  that  (1)  may  be  regarded  as  the 
equation  of  motion  of  any  point  of  the  cross-head  if  a  suitable 
origin  be  selected.  Thus  (1)  is  the  equation  of  motion  of  P 
referred  to  the  origin  Oi,  where  Oi  is  the  middle  point  of  the  up- 
and-down  range  of  motion  of  P. 

185.  If  P,  Fig.  128,  be  a  tracing  point  attached  to  the  vertical 
arm  of  the  cross-head  and  capable  of  describing  a  curve  on  a  uni- 
formly translated  piece  of  smoked  glass,  HK,  then  when  P  de- 
scribes S.H.M.  in  the  vertical  line  OP,  the  curve  NiCTNJP  traced 
on  the  plate  HK  is  a  sinusoid,  for  the  ordinates  on  HK  measured 
with  respect  to  the  median  line  OiN\  are  proportional  to  sin  B  and 
by  hypothesis  the  abscissas  or  horizontal  distances  vary  uniformly. 
If  the  plate  HK  move  with  exactly  the  same  speed  as  the  point  Af , 
the  undistorted  sinusoid  of  Fig.  59  is  described,  whose  equation  is 

V  X 

2/ =  a  sin  — i  =  a  sin—  (1)^ 

a  a 

1  The  student  should  note  that  -  »»  sin  -  is  of  exactly  the  same  shape  as  j/  =■  sin  x, 
a  a 

for  multiplying  both  ordinates  and  abscissas  of  any  curve  by  o  is  merely  constructing 

y  X  . 

the  curve  to  a  different  scale.     However,  „  =  sin  „  is  a  distorted  sinusoid,  for  the 

ordinates  o$y  =  sin  x  are  multiplied  by  3  while  the  abscissas  are  multiplied  only  by  2. 


324         ELEMENTARY  MATHEMATICAL  ANALYSIS      [§186 


where  x  is  the  abscissa  of  any  point  of  the  sinusoid  referred  to  an 

origin  (as  N)  moving  with  the  plate.     If,  however,  the  velocity  of 

the  plate  be  y'  instead  of  v,  then  the  equation  of  the  curve  on  HK, 

referred  to  axes  moving  with  the  plate,  is  of  the  form 

.     x'  ,    px 

=  a  sm  —  ==  a  sm  — 

a  a  a 

x'  =  vx,  and   h 


y  =  asm 
where  v't  = 


a  sin  hx 


(2) 


X 


B 



— ^ 

"■v;;^ 

z 

N 

/ 

\ 

/ 

\^ 

/ 

\ 

r 

, 

O 

/ 

\ 

/ 

\ 

/ 

\ 

/ 

^ 



— . 

^ 

Z 

^    ^ 

^ 

w 

:: 

^^^ 

— / 

.dL. 

— 

-^ 

^ 

s 

/- 



— 

"Y 

5 

\ 

O' 

-^i 

t 

/ 

:s 

^-;- 

!^ 

— 

7^ 

— 

7 

^ 

3 

^ 

=  p/a.  Changing  the  rela- 
tive speed  of  the  wheel  and 
plate  corresponds  to  stretch- 
ing or  contracting  the  sine 
curve  in  the  x  direction. 

186.  Composition  of  Two 
S.H.M.'s  at  Right  Angles. 
We  have  shown  if  a  point  Af, 
moving  uniformly  on  a  circle, 
be  projected  upon  both  the 
X-  and  7-axes,  two  S.H.M.'s 
result.  The  phase  angles  of 
these  two  motions  differ  from 


each  other  by  ^  or  90° 


The 


converse  of  this  fact,  namely 
that  uniform  motion  in  a  cir- 
cle may  be  the  resultant  of 
two    S.H.M.    in    quadrature, 
is  easily  proved,  for  the  two 
equations  of  S.H.M. : 
X  =  a  cos  kt 
y  =  asin  kt 
are  obviously  the  parametric 
equations  of  a  circle.     Hence 
the  theorem: 

Uniform  motion  in  a  circle  may  he  regarded  as  the  resultant  of 
two  S.H.M.'s  of  equal  amplitudes  and  equal  periods  and  differing 
hy  IT  /2  in  phase  angle. 

This  important  truth  is  illustrated  by  Fig.  129.  Let  the  x- 
and  2/-axes  be  divided  proportionally  to  the  trigonometric  sine, 


JFiG.  129.— The  Circle  and  the  Ellipse 
Considered  as  Generated  by  Two 
S.H.M.'s  in  Quadrature. 


§186]  WAVES  325 

as  in  Fig.  59.  Through  the  points  of  division  of  the  two  axes 
draw  lines  perpendicular  to  the  axes,  thus  dividing  the  plane  into 
a  large  number  of  small  rectangles.  Starting  at  the  end  of  one 
of  the  axes,  and  sketching  the  diagonals  of  successive  cornering 
rectangles,  the  circle  ABA'B'  is  drawn. 

If  the  same  construction  be  carried  out  for  the  case  in  which 
the  y-Sixis  is  divided  proportionally  to  b  sin  kt  and  in  which  the 
a;-axis  is  divided  proportionally  to  a  sin  kt,  the  ellipse  AiBiA\B'i 
results.  These  facts  are  merely  a  repetition  of  the  statements 
made  in  §74. 

Exercises 

1.  Draw  a  curve  by  starting  at  the  intersection  of  any  two  lines  of 
Fig.  129,  and  drawing  the  diagonals  of  successive  cornering  rectangles, 
and  write  the  parametric  equations  of  the  curve. 

2.  Find  the  periods  of  the  following  S.H.M. : 

(a)  y  =  S  sin  2t. 

(5)  y  =  10sin(l/2)<. 

(c)  y  =  7  cos  4t. 

{d)  y  =  a  sin  ^-n-t. 

(e)  y  =  a  sin  {lOt  -  tt/S). 

(f)y  =  a  sin  {2t/S  -  27r/5). 

(g)  y  =  a  sin  (bt  +  c). 

3.  Give  the  amplitudes  and  epoch  angles  in  each  of  the  instances 
given  in  example  2. 

4.  The  bob  of  a  second's  pendulum  swings  a  maximum  of  4  cm. 
each  side  of  its  lowest  position.  Considering  the  motion  as  rectilinear 
S.H.M.  write  its  equation  of  motion. ^ 

Write  the  equation  of  motion  of  a  similar  pendulum  which  was 
released  from  the  end  of  its  swing  1/2  second  after  the  first  pendulum 
was  similarly  released. 

6.  A  particle  moves  in  a  straight  line  in  such  a  way  that  its  dis- 
placement from  a  fixed  point  of  the  line  is  given  by  d  =  2  cos^  t. 
Show  that  the  particle  moves  in  S.H.M.,  and  find  the  amplitude 
and  period  of  the  motion. 

6.  A  particle  moves  in  a  vertical  circle  of  radius  2  units  with 
angular  velocity  of  20  radians  per  second.     Counting  time  from  the 

1  The  term  period  is  used  differently  in  the  case  of  a  pendulum  than  in  the 
case  of  S.H.M.  The  time  of  a  awing  is  the  period  of  a  pendulum;  the  time  of 
a  swing-swang  is  the  period  of  a    S.H.M. 


326        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§187 

instant  when  the  particle  is  at  its  lowest  position,  write  the  equa- 
tion of  motion  of  its  projection  (1)  upon  the  vertical  diameter; 
(2)  upon  the  horizontal  diameter;  (3)  upon  the  diameter  bisecting 
the  angle  between  the  horizontal  and  vertical. 

187.  Waves.  The  curve  described  on  the  moving  plate  fl^K 
of  Fig.  128,  if  referred  to  coordinate  axes  moving  with  the  plate , 
is  the  sinusoid  or  sine  curve,  which  for  the  sake  of  greater  generality 
we  shall  suppose  is  of  the  type  (§185,  equation  (2)) 

y  =  asin  hx  (1) 

If,  however,  we  consider  this  curve  as  referred  to  the  fixed  origin 
Oi,  then  the  moving  sinusoid  thus  conceived  is  called  a  simple 
progressive  sinusoidal  wave  or  merely  a  wave.  Under  the  con- 
ditions represented  in  Fig.  128,  it  is  a  wave  progressing  to  the 
right  with  the  uniform  speed  of  the  plate  HK.  At  any  single 
instant,  the  equation  of  the  curve  is: 

y  =  asin  h{x  -  OiN)  (2) 

where  OiN  is  the  distance  that  the  node  N  has  been  translated 
to  the  right  of  the  origin  Oi.  If  V  be  the  uniform  velocity  of 
translation  of  HK,  then: 

OiN  =  Vt  (3)1 

and  the  equation  of  the  wave  is: 

y  =  asin  h(x  —  Vt) 
or, 

y  =  a  sin  (hx  —  kt)  (4) 

if  k  be  put  for  hV,  so  that 

V  =  I  (5) 

Because  of  the  presence  of  the  variable  t,  this  is  not  the  equation 
of  a  fixed  sinusoid,  but  of  a  moving  sinusoid  or  wave. 

Applying  the  same  terms  used  for  S.H.M.,  the  expression 
{hx  —  kt)  is  the  phase  angle,  the  expression  (  +  kt)  is  the  epoch 
angle  and  a  is  the  amplitude  of  the  wave.     See  Fig.  130a  and  c. 

The  expression  {hx  —  kt)  is  a  linear  function  of  the  variables 

1  In  what  follows,  t  is  not  the  time  elapsed  since  M,  Fig.  128,  was  at  A,  as  used  in 
§184,  but  is  the  elapsed  time  since  N  was  at  Oi.  These  values  of  t  differ  by  the  time 
of  half  a  revolution  or  by  ir/k. 


§188] 


WAVES 


327 


X  and  t.    The  sine  or  cosine  of  this  function  is  called  a  simple 
harmonic  function  of  x  and  i. 

188.  Wave  Length.    Since  the  period  of  the  sine  is  27r,  if  i 
remain  constant  and  the  expression  hx  be  changed  by  the  amount 


^::?'<:^S^-y^y 


X  X  X  > 


Fig.  130. — Wave  Forms,   (a)   of  Different  Amplitude;  (5)   of  Different 
Wave  Lengths;  (c)  of  Different  Phase  or  Epoch  Angles. 

2'n',  the  curve  (4)  is  translated  to  the  left  or  right  an  amount  such 
that  trough  coincides  with  trough  and  crest  coincides  with  crest, 
and  the  curve  in  its  second  position  coincides  with  the  curve  in 


328        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§189 

its  first  position.  Call  x^  the  abscissa  of  any  point  of  the  curve 
in  its  second  position  whose  original  abscissa  was  Xi.     Then: 

hxi  —  hxi  =  2ir 
or: 

^2  —  Xi  =  27r//i 
Calling  the  distance  x^  —  Xi  =  L,  we  have: 

L  =  2Trlh  (1) 

L  is  called  the  wave  length.  It  is  the  distance  from  any  crest 
to  the  next  crest  or  from  any  trough  to  the  next  trough  or  from 
any  node  to  the  second  succeeding  node,  or  from,  any  point  of 
the  wave  to  the  next  similar  point.     See  Fig.  1306. 

The  wave  length  can  also  be  determined  in  the  following  manner: 
The  wave  length  of 

y  =  sinx  (2) 

is  obviously  27r,  the  length  of  the  period  of  the  sine.  The  sine 
curve 

y  =  sin  hx  (3) 

can  be  made  from  the  above  by  multiplying  the  abscissas  of  all 

1  .    2t 

points  by  r.     Therefore  the  wave  length  of  the  latter  is  -v- .     The 

wave  length  of 

y  =  sin  (hx  —  kt)  (4) 

must  also  be  the  same  as  that  of  (3),  since  the  effect  of  the  term 
kt  is  merely  to  translate  the  curve  as  a  whole  a  certain  distance 
to  the  right. 

189.  Period  or  Periodic  Time.  If  we  fix  our  attention  upon 
any  constant  value  of  x,  and  if  kt  in  (4)  above  be  permitted  to 
change  by  the  amount  27r,  then  since  the  period  of  the  sine  is 
27r,  the  curves  at  the  two  instances  of  time  mentioned  must  coin- 
cide.    Calling  the  two  values  of  t,  ti  and  ti,  we  have  by  hypothesis 

ktz  —  kti  =  27r 
Writing: 

t2-ti    =    T 

we  find 

T  =  27r/k  (1). 

The  expression  T  is  called  the  periodic  time  or  period  of  the 
wave.     It  is  the  length  of  time  required  for  the  wave  to  move  one 


§190]  WAVES  329 

wave  length,  or  the  length  of  time  that  elapses  until  trough  again 
coincides  with  trough,  etc.  To  contrast  wave  length  and  period, 
think  of  a  person  in  a  boat  anchored  at  a  fixed  point  in  a  lake.  The 
time  that  the  person  must  wait  at  that  fixed  point  {x  constant) 
for  crest  to  follow  crest  is  the  periodic  time.  The  wave  length 
is  the  distance  he  observes  between  crests  at  a  given  instant  of 
time  (^constant). 

190.  Velocity  or  Rate  of  Propagation.  The  rate  of  movement 
V  of  the  sinusoid  on  the  plate  HK,  Fig.  128,  is  shown  by  equa- 
tion (5),  §187,  to  be  k/h  units  of  length  per  second.  This  is 
called  the  velocity  of  the  wave  or  the  velocity  of  propagation. 
The  equation  of  the  wave  may  be  written: 

y  —  a  sin  h{x  —  Vt) 

From  equations  (1)  §188  and  (1)  §189  we  may  write 


whence 


k 

Since  F  =  v)  we  have: 

h 


k    _   L 

h   ~    T 


V  =  ^  (1) 

This  equation  is  obvious  from  general  considerations,  for  the 
wave  moves  forward  a  wave  length  L  in  time  T,  hence  the  speed 

of  the  wave  must  be  ^• 

191.  Frequency.  The  number  of  periods  per  unit  of  time  is 
called  the  frequency  of  the  wave.  Hence,  if  N  represent  the 
frequency  of  the  wave, 

«  =  T   =   2^  (2) 

There  is  no  name  given  to  the  reciprocal  of  the  wave  length. 

192.  L  and  T  Equation  of  a  Wave.    If  we  solve  equations  (1) 


330        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§193 

§188  and   (1)    §189  for  h  and  k   respectively,   and    substitute 
these  values  of  h  and  k  in  the  equation 

2/  =  a  sin  {hx  —  kt) 
we  obtain 


[-^] 


y  =  asin27r  j^    -    ^  (1) 

From  this  form  it  is  seen  that  the  argument  of  the  sine  increases 
by  27r  when  either  x  increases  by  an  amount  L  or  when  t  increases 
by  the  amount  T.  By  use  of  (1),  §190,  the  last  equation  may 
also  be  written: 

y  =  asin^(x-Vt)  (2) 

193.  Phase,  Epoch,  Lead.     Consider  the  two  waves: 

?/  =  a  sin  Y  (^  ~  ^0  (1) 

2t 
2/  =  a  sin  -v-(a;  -  Vt  -  E)  (2) 

The  amplitudes,  the  wave  lengths  and  the  velocities  are  the 
same  in  each,  but  the  second  wave  is  in  advance  of  the  first  by 
the  amount  E  (measured  in  linear  units),  for  the  second  equation 
can  be  obtained  from  the  first  by  substituting  (x  —  E)  for  x,  which 
translates  the  curve  the  amount  E  in  the  OX  direction.  In  this  case 
E  is  called  the  lead  (or  the  lag  if  negative)  of  the  second  wave 
compared  with  the  first. 

The  lead  is  a  linear  magnitude  measured  in  centimeters,  inches, 
feet,  etc.  The  epoch  angle  is  measured  in  radians.  In  the 
present  case  the  epoch  angle  of  (2)  is  2'iriVt  +  E)  jL. 

The  terms  phase  and  epoch  are  sometimes  used  to  designate 
the  time,  or,  more  accurately,  the  fractional  amount  of  the  period 
required  to  describe  the  phase  angle  and  epoch  angle  respectively. 
In  this  use,  the  phase  is  the  fractional  part  of  the  period  that  has 
elapsed  since  the  moving  point  last  passed  through  the  middle  point 
of  its  simple  harmonic  motion  in  the  direction  reckoned  as  positive. 
See  Fig.  130c. 

The  tidal  wave  in  mid  ocean,  the  ripples  on  a  water  surface, 
the  wave  sent  along  a  rope  that  is  rapidly  shaken  by  the  hand, 
are  illustrations  of  progressive  waves  of  the  type  discussed  above. 


§193]  WAVES  331 

Sound  waves  also  belong  to  this  class  if  the  alternate  condensations 
and  rarefactions  of  the  medium  be  graphically  represented  by 
ordinates.  The  ordinary  progressive  waves  observed  upon  a  lake 
or  the  sea  are  not,  however,  progressive  waves  of  this  type.  The 
surface  of  the  water  in  this  case  is  not  sinusoidal  in  form,  but 
is  represented  by  another  class  of  curves  known  in  mathematics 
as  trochoids. 

Exercises 

1.  Derive  the  amplitude,  the  wave  length,  the  periodic  time,  the 
velocity  of  propagation  of  the  following  waves: 

y  =  a  sin  (2x  —  30- 

2/  =  5  sin  (0.75a;  -  10000- 

y  =  lOsin  (I  -  1) 

y  =  50  sin  y(a:  —  30- 

y  =  100sin||(x  -  20f  -4). 

y  =  100  sin  {5x  +  40. 

y  =  0.025  sin  —(x  + 1/3). 

2.  Write  the  equation  of  a  progressive  sinusoidal  wave  whose  height 
is  5  feet,  length  40  feet  and  velocity  4  miles  per  hoi^r. 

3.  Write  the  equation  of  a  wave  of  wave  length  10  meters,  height 
1  meter  and  velocity  of  propagation  3.5  miles  per  hour.  (Note: 
1  mile  =  1.609  kilometers.) 

4.  Sound  waves  of  all  wave  lengths  travel  in  still  air  at  70°  F.  with 
a  velocity  of  1130  feet  per  second.  Find  the  wave  length  of  sound 
waves  of  frequencies  256,  128,  600  per  second. 

6.  The  lowest  note  recognizable  as  a  musical  tone  was  found  by 
Helmholtz  to  possess  about  40  vibrations  per  second.  The  highest 
note  distinguishable  by  an  ordinary  ear  possesses  about  20,000  vibra- 
tions per  second.  If  the  velocity  of  sound  in  air  be  1130  feet  per 
second,  find  the  wave  length  in  each  of  these  limiting  cases,  and  write 
the  equation  of  the  waves  if  the  amplitude  be  represented  by  the 
symbol  o. 


332        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§194 

194.  Stationary  Waves.  The  form  of  a  violin  string  during  its 
free  vibration  is  sinusoidal,  but  the  nodes,  crests,  troughs,  etc., 
are  stationary  and  not  progressive  as  in  the  case  of  the  waves 
just  discussed,  and  is  therefore  called  a  stationary  wave.  The 
water  in  a  basin  or  even  in  a  large  pond  or  lake  is  also  capable  of 
vibrating  in  this  way.  Fig.  131  may  be  used  to  illustrate  the 
stationary  waves  of  this  type,  either  of  a  musical  string  or  of  the 
water  surface  of  a  lake,  but  in  the  case  of  a  vibrating  string,  the 
ends  must  be  supposed  to  be  fastened  at  the  points  0  and  N. 
The  shores  of  the  lake  may  be  taken  at  I  and  K  or  at  /  and  H, 
etc.  As  is  well  known,  such  bodies  are  capable  of  vibrating  in 
segments  so  that  the  number  of  nodes  may  be  large.     This 


Fig.   131. — A  Stationary  Wave. 


explains  the  "harmonics"  of  a  vibrating  violin  string  and  the 
various  modes  in  which  stationary  waves  may  exist  on  a  water 
surface.  A  stationary  wave  on  the  surface  of  a  lake  or  pond  is 
known  as  a  seiche,  and  was  first  noted  and  studied  on  Lake 
Geneva,  Switzerland.  The  amplitudes  of  seiches  are  usually 
small,  and  must  be  studied  by  means  of  recording  instruments 
so  set  up  that  the  influence  of  progressive  waves  is  eliminated. 
The  maximum  seiche  recorded  on  Lake  Geneva  was  about  6  feet, 
although  the  ordinary  amplitude  is  only  a  few  centimeters. 

The  equation  of  a  stationary  wave  may  be  found  by  adding  the 
ordinates  of  a  progressive  wave: 


y  =  asin  (hx  —  kt) 


(1) 


traveling  to  the  right  (/:  >  0),  to  the  ordinates  of  a  progressive 
wave: 

2/  =  a  sin  {hx  +  kt)  (2) 

traveling  to  the  left. 


§194]  WAVES  333 

Expanding  the  right  members  of  (1)  and  (2)  by  the  addition 
formula  for  the  sine,  and  adding: 

y  =  2a  cos  kt  sin  hx  (3) 

or  in  terms  of  L  and  T 

y  =  2a  cos  (^^  sin  (^^  (4) 

In  Fig.  131,  the  origin  is  at  0  and  the  X-axis  is  the  line  of  nodes 
ONX.  If  we  look  upon  2a  cos  kt  as  the  variable  amplitude  of 
the  sinusoid 

y  =  sin  hx 
we  note  that  the  nodes,  etc.,  of  the  sinusoid  remain  stationary, 
but  that  the  amphtude  2a  cos  kt  changes  as  time  goes  on.  When 
t  =  0,  the  sine  curve  has  amplitude  2a  and  wave  length  27r/A. 
When  t  =  TT  12k  or  r/4  the  sinusoid  is  reduced  to  the  straight  line 
y  =  0.     When  t  =  tt  jk  or  T 12  the  curve  is  the  sinusoid: 

y  =  —  2a  sin  hx 
which  has  a  trough  where  the  initial  form  had  a  crest,  and  vice 
versa. 

Exercises 

In  the  following  exercises  the  height  of  the  wave  means  the  maxi- 
mum rise  above  the  line  of  nodes.  When  a  seiche  is  uninodal,  the 
shores  of  the  lake  correspond  to  the  points  /  and  K,  Fig,  131.  When 
a  seiche  is  binodal,  the  points  /  and  H  are  at  the  lake  shore. 

1.  From  the  equation  of  a  stationary  wave  in  the  form  y  = 
2a  sin  2Trx/L  cos  2Trt/T,  show  that  K,  Fig.  131,  is  at  its  lowest  depth 
for  t  =  T/2,  3r/2,  5T/2,   .    .    . 

2.  Henry  observed  a  fifteen-hour  uninodal  seiche  in  Lake  Erie,  which 
was  396  kilometers  in  length.  Write  the  equation  of  the  principal 
or  uninodal  stationary  wave  if  the  amplitude  of  the  seiche  was  15  cm. 

3.  A  small  pond  111  meters  in  length  was  observed  by  Endros  to 
have  a  uninodal  seiche  of  period  fourteen  seconds.  Write  the  equa- 
tion of  the  stationary  wave  if  the  amplitude  be  a. 

4.  Forel  reports  that  the  uninodal  longitudinal  seiche  of  Lake  Geneva 
has  a  period  of  seventy-three  minutes  and  that  the  binodal  seiche  has 
a  period  of  thirty-five  and  one-half  minutes.  The  transverse  seiche 
has  a  period  of  ten  minutes  for  the  uninodal  and  five  minutes  for 
the  binodal.     The  longitudinal  and  transverse  axes  of  the  lake  are 


334        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§195 

45  miles  and  5  miles  respectively.  Write  the  equation  of  these 
different  seiches. 

5.  A  standing  wave  or  uninodal  seiche  exists  on  Lake  Mendota 
of  period  twenty-two  minutes.  If  the  maximum  height  is  8  inches 
and  the  distance  across  the  lake  is  6  miles,  write  the  equation  of 
the  seiche. 

195.  Compound    Harmonic    Motion    and    Compound  Waves. 

The  addition  of  two  or  more  simple  harmonic  functions  of  dif- 
ferent periods  gives  rise  to  compound  harmonic  motion.     Thus : 

y  =  a  sin  kt  -{-  b  sin  3  kt 

corresponds  to  the  superposition  of  a  S.H.M.  of  period  27r/3fc 
and  amplitude  b  upon  a  fundamental  S.H.M.  of  period  2Tr/k 
and  amplitude  a.  To  compound  motions  of  this  type,  there  cor- 
respond compound  waves  of  various  sorts,  such  as  a  fundamental 
sound  wave  with  overtones,  or  tidal  waves  in  restricted  bays  or 
harbors.     The  graphs  of  the  curves: 


Fig.  132. — The  Curves  y  =  sin  x,  y  =  sin  3x  and  the  Compound  Curve 
2/  =  sin  X  +  sin  3x. 

y  =  sinx  -\-  sin  2x 
y  =  sinx  -j-  sin  Sx 
y  =  sinx  -{-  sin  2x  +  sin  Sx 

should  be  constructed  by  the  student.  They  may  be  drawn  by 
adding  the  ordinates  of  the  various  sinusoids  constructed  on  the 
same  axis,  as  in  Fig.  132.  To  compound  the  curves,  first  draw 
the  component  curves,  say  y  =  smx  and  y  —  sin  3a;  of  Fig.  132. 
Then  use  the  edge  of  a  piece  of  paper  divided  proportionally 
to  sin  X  (that  is,  like  the  scale  OB,  Fig.  132)  and  use  this  as  a 
scale  by  means  of  which  the  successive  ordinates  of  a  given  x 


§195] 


WAVES 


335 


may  be  added.  For  example,  to  locate  the  point  on  the  composite 
curve  corresponding  to  the  abscissa  OD,  Fig.  132,  we  must  add 
DP  and  DQ.  Hence  place  vertically  at  P  the  lower  end  of  the 
paper  scale  just  mentioned.  The  sixth  scale  division  above  P 
on  this  scale  will  then  locate  the  required  point  M  of  the  composite 
scale. 


Octave 


Fifth 


Fig.  133.— The  Curves  (a)  y  =  sinx  +  sin  (2a;  +  27rn/16)  and  (6)  y  = 
sin  2x  +  sin(3x  +  2vn/lQ),  for  n  =  0,  1,  2,  .  .  .  15.  {From  Thomson 
and  Tail.) 

In  Fig.  133  the  curves: 

y  =  sin  a;  +  sin  {2x  +  Sttw  /16) 
y  =  sin  2a;  +  sin  (3a;  +  27rn/16) 
are  shown  for  values  of  n  =  0,  1,  2,  .    .    .,  15  in  succession — 
that  is,  for  successive  phase  differences  corresponding  to  one-six- 
teenth of  the  wave  length  of  the  fundamental  y  =  sin  x. 

Wave  forms  compounded  from  the  odd  harmonics  only  are 
especially  important,  as  alternating-current  curves  are  of  this  type. 
See  Fig.  134. 


336        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§197 

196.  Harmonic  Analysis.  Fourier  showed  in  1822  in  his 
''Analytical  Theory  of  Heat"  that  a  periodic  single- valued 
function,  say  y  =  f{x),  under  certain  conditions  of  continuity, 
can  be  represented  by  the  sum  of  a  series  of  sines  and  cosines  of 
the  multiple  angles  of  the  form: 

2/  =  «o  +  Oi  cos  X  +  02  cos  2x  +  cia  cos  3a;  +  .    .    . 
+  hi  sin  0^  +  62  sin  2a;  +  63  sin  3a;  +  .    .    . 
This  means,  for  example,  that  it  is  always  possible  to  represent 
the  complex  tidal  wave  in  a  harbor,  by  means  of  the  sum  of  a 


_J 

___ 

/ 

/' 

\ 

0- 

/ 

\ 

f 

N. 

/ 

\ 

s 

0 

/ 

, 

5 

;    1 

3    1 

2     1 

i  1 

6H 

^  2 
s 

0    2 

2    2 

4    2 

5    2 

8    i 

0    3 

9  '. 

V' 

\ 

/ 

\ 

/ 

V 

y 

en 

\ 

^ 

/ 

Fig.   134. — An  Alternating  Current  Curve. 

present. 


Only  Odd   Harmonics  are 


number  of  simple  waves  or  harmonics.  The  term  harmonic 
analysis  is  given  to  the  process  of  determining  these  sinusoidal 
components  of  a  compound  periodic  curve.  In  §195  we 
have  performed  the  direct  operation  of  finding  the  compound 
curve  when  the  component  harmonics  are  given.  The  inverse 
operation  of  finding  the  components  when  the  compound  curve 
is  given  is  much  more  difficult,  and  its  discussion  must  be  post- 
poned to  a  later  course. 

197.*  Test  for  a  Sinusoidal  Function.  Squared  paper,  known 
as  semi-sinusoidal  paper,  has  been  prepared  (see  Fig.  135) 
with  the  horizontal  scale  divided  proportionally  to  sin  x  and  the 
vertical  scale  divided  uniformly.  The  divisions  are  precisely 
the  same  as  those  in  Fig.  59,  except  that  the  number  of  divisions 


11971 


WAVES 


337 


is  greatly  increased.  On  this  paper,  the  sine  curve  is  represented 
by  a  straight  line  drawn  diagonally  across  the  paper.  Since  the 
sine  curve  appears  as  a  straight  line  on  this  paper  (just  as  the 
logarithmic  curve  appears  as  a  straight  line  on  semi-logarithmic 
paper)  it  is  easy  to  test  whether  or  not  observed  periodic  data 
follow  the  law  expressed  by  the  sine  curve.  Thus  the  times  of 
sunrise  at  Boston,  Massachusetts,  for  the  first  day  of  each  month 
have  been  plotted  upon  the  sheet  shown  in  Fig.  135.     The  points 


A.M.8:00 
40 
J 
20 

7:00 

40 

ao 

6:00 
40 
20 

5:00 
40 
20 

4:00 


1 

"s 

F 

S- 

^ 

^ 

M 

^^ 

N 

^**^^ 

%>-_. 

o 

^ 

A 

^ 

■«»_ 

^s 

^^ 

^ 

"^ 

^^ 

.     A 

mH 

K* 

^S 

s 

•^ 

8:00  A.M. 
40 
20 
7:00 
40 

20   • 
6:00 
40 
20 
5:00 
40 
J 
20 
4:00 


Fig.   135. — Time      of      Sunrise      on     the      First      Day      of      Each 
Month  at  Boston,  Mass.,  Graphed  Upon  Semi-sinusoidal  Paper. 


corresponding  to  the  various  dates  do  not  form  a  straight  line, 
although  it  is  obvious  that  the  sine  curve  is  a  first  approximation 
to  the  proper  curve. 

The  times  of  sunrise  plotted  in  Fig.   135  are  given  in  exercise 
(1)  below. 

Exercises 

1.  The  times  of  sunset  and  sunrise  at  Boston,  Massachusetts,  for 
the  first  day  of  each  month  are  as  follows : 

JFMAMJJASOND 
Sunset       4:37  5:13  5:49  6:25  6:59  7:29  7:40  7:21  6:37  5:44  4:55  4:28 
Sunrise      7:30  7:15  6:38  5:44  4:56  4:26  4:25  4:51  5:23  5:56  6:31  7:09 
Length  of  day  

Graph  the  times  of  sunset  upon  semi-sinusoidal  paper. 

Note  :    The  earliest  sunset  tabulated  is  December  1,  which  should 
be  used  for  the  date  of  the  trough  of  the  wave. 

22 


338        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§198 

2.  Determine  the  length  of  the  day  from  the  data  of  exercise  1,  and 
graph  the  same  upon  semi-sinusoidal  paper. 

3.  Determine  whether  the  curves  of  Fig.  136  are  sinusoidal  or  not. 

198.  Connecting  Rod  Motion.  If  one  end  of  a  straight  line  B 
be  required  to  move  on  a  circle  while  the  other  end  of  the  line  A 
moves  on  a  straight  line  passing  through  the  center  of  the  circle, 


JFMAMJ^JASONDJ 

Max. 
2  P.M. 
9  P.M. 

7  A.M 

Max. 

2  P.M. 

9  P.M. 
7A,M. 

Min. 

h   ^ 

N 

\ 

/ 

n\ 

/y 

\\ 

// 

/s^>>. 

\ 

// 

YT^ 

u    \ 

\ 

/, 

V 

>. 

\ 

1 

/ 

\ 

\ 

h 

-> 

v\ 

// 

/a 

ps 

\ 

// 

f 

\V 

— 1 

// 

V 

/ 

\ 

/ 

/ 

5 

0° 

. 

n° 

^ 

n° 

—6^^ 

^ 

^/ 

.  ^^ 

/ 

v^ 

""^ 

20° 

.  ^ 

/ 

\^ 

V^    "• 

^ 

,  ^< 

Min. 

\ 

10^ 

\ 

^ 

1 

s 

Fig.   136. — Daily  Temperatures  throughout  the  Year  at  Madison,  Wis., 
Average  of  many  years. 

the  resulting  motion  is  known  as  connecting  rod  motion.  The 
connecting  rod  of  a  steam  engine  has  this  motion,  as  the  end  at- 
tached to  the  crank  travels  in  a  circle  while  the  end  attached  to 
the  piston  travels  in  a  straight  line.  The  motion  of  the  end  A 
of  the  connecting  rod  is  approximately  S.H.M.  The  approxi- 
mation is  very  close  if  the  connecting  rod  be  very  long  in  compari- 
son with  the  diameter  of  the  circle. 
A  second  approximation  to  the  motion  of  the  point  A  can  be 


1198] 


WAVES 


339 


shown  to  introduce  the  second  harmonic  or  octave  of  the  funda- 
mental. In  Fig.  137,  let  the  radius  of  the  circle  be  a  and  the 
length  of  the  connecting  rod  be  I.  The  length  of  the  stroke  MN 
is  2a,  and  the  origin  may  conveniently  be  taken  at  the  mid  point 
of  the  stroke,  0.  When  B  was  at  H,  A  was  at  M  and  when  B 
was  at  K,  A  was  at  N.  Then  MH  =  NK  =  I,  and  OC  =  I. 
Now 

X  =  CA  -  CO  =  CA  -  I  =  CD  -^  DA  -  I 


But 


and 


Hence: 


CD  =  a  cos  6 


DA  =    VZ2  -  BD^ 


=    <P 


a^  sin^  9 


=  a  cos  9  +  i  Vl  -  {a^ll^)  sin^  9  -  I 


(1) 
(2) 


(3) 


(4) 


S.HJ4,4-=O0 

O.RJVt.4=  9 
OMM.  -i=  6 
O.K.M.-I--  3 

Fig.   137. — Connecting  Rod  Motion. 


Approximating   the  radical   by    §111   (Vl  —  a;  =  1  —  a;/2)    we 
obtain: 

.   ,    ,  /,       a2  sin2  e^        - 
a;  =  a  cos  0  +  Z  ( 1 2/2 — )  ~  ^ 

But  sin2  e  =  {\  -  cos  26)12,  hence: 


(5) 


x  =  a  cos 


+  ^  cos  26 


42 


(6) 


which  is  approximately  true  as  long  as  I  is  much  greater  than  a. 

It  is  seen  from  the  above  result  that  the  second  approximation 

to  connecting  rod  motion  contains  as  overtone  the  octave  or 

second  harmonic,  -j^  cos  26,  in  addition  to  the  first  or  fundamental 

harmonic  a  cos  6. 


340         ELEMENTARY  MATHEMATICAL  ANALYSIS       §198 

Exercises 

1.  Draw  the  curve  corresponding  to  equation  (5)  above  if  a  =  1.15 
inches,  and  I  =  3  inches.     - 

2.  The  motion  of  a  slide  valve  is  given  by  an  equation  of  the  form: 

y  =  ai  sin  (0  +  e)  +  ^2  sin  (2d  +  90°). 
Draw  the  curve  if  ai  =  100,  az  =  25,  €  =  40°. 

3.  Graph: 

y  =  5sm{d  +  30°)  +  2  sin  (26  +  90°). 

4.  Graph: 

y  =  sinx  +  (1/3)  sin  Sx  +  (1/5)  sin  5x. 


CHAPTER  XI 
COMPLEX  NUMBERS 

199.  Scales  of  Numbers.  To  measure  any  magnitude,  we 
apply  a  unit  of  measure  and  then  express  the  result  in  terms  of 
numbers.  Thus,  to  measure  the  volume  of  the  liquid  in  a  cask 
we  may  draw  off  the  liquid,  a  measure  full  at  a  time,  in  a  gallon 
measure,  and  conclude,  for  example,  that  the  number  of  gallons 
is  12^.  In  this  case  the  number  12^  is  taken  from  the  arith- 
metical scale  of  numbers,  0,  1,  2,  3,  4,  .  .  .  If  we  desire  to  meas- 
ure the  height  of  a  stake  above  the  ground,  we  may  apply  a 
foot-rule  and  say,  for  example,  that  the  height  in  inches  above  the 
ground  is  12^,  or,  if  the  positive  sign  indicates  height  above  the 
ground,  we  may  say  that  the  height  in  inches  is  -f-  12|.  In 
this  latter  case  the  number  4-12^  has  been  selected  from  the 
algebraic  scale  of  numbers  .  .  .  —  4,  —  3,  —  2,  —  1,  0,  -|-  1, 
-h  2,  -F  3,  -h  4,   .    .    . 

The  scale  of  numbers  which  must  be  used  to  express  the  value  of  a 
magnitude  depends  entirely  upon  the  nature  of  the  magnitude.  The 
attempt  to  express  certain  magnitudes  by  means  of  numbers  taken 
from  the  algebraic  scale  may  sometimes  lead,  as  every  student  of 
algebra  knows,  to  meaningless  absurdities.  Thus  a  problem  involving 
the  number  of  sheep  in  a  pen,  or  the  number  of  marbles  in  a  box,  or 
the  number  of  gallons  in  a  cask,  cannot  lead  to  a  negative  result,  for 
the  magnitudes  just  named  are  arithmetical  quantities  and  their  meas- 
urement leads  to  a  number  taken  from  the  arithmetical  scale.  The 
absurdity  that  sometimes  appears  in  results  to  problems  concerning 
these  magnitudes  is  due  to  the  fact  that  one  attempts  to  apply  the 
notion  of  algebraic  number  to  a  magnitude  that  does  not  permit  of  it. 
Science  deals  with  a  great  many  different  kinds  of  magnitudes,  the 
measurement  of  some  of  which  leads  to  arithmetical  numbers  while  the 
measurement  of  others  leads  to  algebraic  numbers;  the  remarkable 
fact  is  that  two  different  number  scales  serve  adequately  to  express 
magnitudes  of  so  many  different  sorts.  The  magnitudes  of  science 
are  so  various  in  kind  that  one  might  reasonably  expect  that  the  num- 
ber of  number  systems  required  in  the  mathematics  of  these  sciences 
would  be  very  great. 

341 


342        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§199 

The  arithmetical  scale,  which  includes  integral  and  fractional 
numbers,  is  itself  more  general  than  is  required  for  the  expression  of 
some  magnitudes.  For  some  magnitudes /rations  are  absurd — quite 
as  absurd,  in  fact,  as  negative  values  are  for  other  magnitudes.  Thus 
the  number  of  teeth  on  a  gear  wheel  cannot  be  a  fraction.  The 
solution  of  the  following  problem  illustrates  this:  "How  many  teeth 
must  be  cut  on  a  pinion  so  that  when  driven  by  a  spur  gear  wheel 
of  fifty-two  teeth  it  will  revolve  exactly  five  times  as  fast  as  the  gear 
wheel?" 

The  arithmetical  scale  is  used  when  we  enumerate  the  number  of 
gallons  in  a  cask  and  say:  0,  1,  2,  3,  .  .  .  If  we  observe  3  gal- 
lons in  the  cask,  and  then  remove  one,  we  note  those  remaining  and 
say  two;  we  may  remove  another  gallon  and  say  one;  we  may  remove 
the  last  gallon  and  say  zero;  but  now  the  magnitude  has  come  to  an 
end — no  more  liquid  may  be  removed. 

Another  conception  of  numerical  magnitude  is  used  when  we  meas- 
ure in  inches  the  height  of  a  stake  above  the  ground  and  say  three 
We  may  drive  the  stake  down  an  inch  and  say  two;  we  may  drive  the 
stake  another  inch  and  say  one;  we  may  drive  the  stake  another  inch 
and  say  zero,  or  "level  with  the  ground;"  but,  unlike  the  case  of  the 
gallons  in  the  cask,  we  need  not  stop  but  may  drive  the  stake  another 
inch  and  say  one  below  the  ground,  or,  for  brevity,  minus  one;  and  so  on 
indefinitely,  but  always  prefixing  "minus"  or  "below  the  ground" 
or  some  expression  that  will  show  the  relative  position  with  respect  to 
the  zero  of  the  scale.  In  this  case  we  have  made  use  of  the  algebraic 
scale  of  numbers. 

Likewise,  in  estimating  time,  there  is  no  zero  in  the  sense  of  the 
gallons  in  the  cask  from  which  to  reckon;  we  cannot  conceive  of  an 
event  so  far  past  that  no  other  event  preceded  it;  we  therefore  select  a 
standard  event,  and  measure  the  time  of  other  events  with  reference 
to  the  lapse  before  or  after  that;  that  is,  we  measure  time  by  means  of 
the  algebraic  scale;  the  symbols  "B.C."  or  "  A.D."  could  quite  as  well 
be  replaced  by  the  symbols  "minus"  and  "plus"  of  the  algebraic  scale. 
The  zero  used  is  an  arbitrary  one  and  the  magnitude  exists  in  reference 
to  it  in  two  opposite  senses,  future  and  past,  or,  as  is  said  in  algebra, 
positive  and  negative.  We  are  likewise  obliged  to  recognize  quantity 
as  extending  in  two  opposite  senses  from  zero  in  the  attempt  to  measure 
many  other  things;  in  locating  points  along  an  east  and  west  line,  no 
point  is  so  far  west  that  there  are  no  other  points  west  of  it,  hence  the 
points  could  not  be  located  on  an  arithmetical  scale;  the  same  in 
measuring  force,  which  may  be  attractive  or  repulsive;  or  motion,  which 
may  be  toward  or  from,  or  rotation,  which  may  be  clockwise  or  anti- 


§199] 


COMPLEX  NUMBERS 


343 


dockwise,  etc.  Because  of  the  necessity  of  measuring  such  magni- 
tudes, our  notion  of  algebraic  number  has  arisen. 

Many  of  the  magnitudes  considered  in  science  are  completely  ex- 
pressed by  means  of  arithmetical  numbers  only;  for  example,  such 
magnitudes  as  density  or  specific  gravity;  temperature;^  electrical  re- 
sistance; quantity  of  energy;  such  as  ergs,  joules  or  foot-pounds; 
power,  such  as  horse  power,  kilowatts,  etc.  All  of  the  magnitudes 
just  mentioned  are  scalar,  as  it  is  called;  that  is,  they  exist  in  one 
sense  only — not  in  one  St'nse  and  also  in  the  opposite  sense,  as  do 
forces,  velocities,  distances,  as  explained  above.  The  arithmetical 
scale  of  numbers  is  therefore  ample  for  their  expression. 

The  distinction,  then,  between  an  algebraic  number  and  an  arith- 
metical number  is  the  notion  of  sense  which  must  always  be  associated 
with  any  algebraic  number.  Thus  an  algebraic  number  not  only 
answers  the  question  ^'how  many"  but  also  affirms  the  sense  in  which 
that  number  is  to  be  understood;  thus  the  algebraic  number 
+  12 1,  if  arising  in  the  measurement  of  angular  magnitude,  refers  to 
an  angular  magnitude  of  12|  units  (degrees,  or  radians,  etc.)  taken 
in  the  sense  defined  as  positive  rotation. 


Exercises 

Of  the  following  magnitudes,  state  which  may  and  which  may  not 
be  represented  adequately  by  an  prithmetical  number: 


1.  10  volts. 

2.  15  calories. 

3.  25  dynes. 

4.  2  kilograms. 

5  20  miles  per  hour. 

6.  4  acre-feet. 

7.  180  revolutions  per  second. 

8.  6-cylinder  (engine). 

9.  3 'atmospheres. 

10.  20  light-years. 

11.  27°  visual  angle. 

12.  Atomic  weight  of  oxygen. 

13.  28  amperes. 

14.  7^  pounds  per  gallon. 


15.  10°  centigrade. 

16.  272°  absolute  temperature. 

17.  16 feet  per  second  (velocity). 

18.  32.2  feet  per  second  per  sec- 
ond (acceleration). 

19.  200  gallons  per  minute. 

20.  20  pounds  per  square  inch. 

21.  50  horse  power. 

22.  1.15  radians  per  second. 

23.  30°  latitude. 

24.  14°  angle  of  depression. 

25.  18  cents  per  gallon. 

26.  60  beats  per  minute. 

27.  5360  feet  above  sea  level. 

28.  312  B.C. 

1  Temperature  is  an   arithmetical  quantity,  since  there   is  an  absolute  zero  of 
temperature.    Temperature  does  not  exist  in  two  opposite  senses,  but  in  a  single 


344        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§201 

200.  Algebraic  Number  Not  the  Most  General  Sort.  Algebraic 
numbers,  although  more  general  than  arithmetical  numbers,  are 
themselves  quite  restricted.  Sir  William  Hamilton,  in  order  to 
emphasize  the  restricted  character  of  an  algebraic  number,  called 
algebra  the  "science  of  pure  time."  That  is,  algebraic  magnitude 
exists  in  the  same  restricted  sense  that  time  exists — because  if  we 
fix  our  attention  upon  any  event,  time  exists  in  one  sense  (future) 
and  in  the  exactly  opposite  sense  (past),  but  in  no  other  sense  at  all. 
Likewise,  with  the  algebraic  numbers,  each  number  corresponds  to 
a  point  of  the  algebraic  scale  (see  §1);  but  for  points  not  on 
the  scale,  or  for  points  sidewise  to  the  same,  there  corresponds  no 
algebraic  number.  This  is  a  way  of  saying  that  the  algebraic 
scale  is  one-dimensional;  Sir  William  Hamilton  desired  to  emphasize 
this  restriction  by  speaking  of  the  "science  of  pure  time,"  for  it 
is  of  the  very  essence  of  the  notion  of  time  that  it  has  one  dimension 
and  one  dimension  only.  It  is  thus  seen  that  there  is  an  opportu- 
nity of  enlarging  our  conception  of  number  if  we  can  remove  the 
restriction  of  one  dimension — that  is,  if  we  can  get  out  of  the  line 
of  the  algebraic  scale  and  set  up  a  number  system  such  that  one 
number  of  the  system  will  correspond,  for  example,  to  each  point  of 
a  plane,  and  such  that  one  point  of  the  plane  will  correspond  to 
each  number  of  the  system.  We  will  seek  therefore  an  exten- 
sion or  generalization  of  the  number  system  of  algebra  that  will 
enable  us  to  consider,  along  with  the  points  of  the  algebraic  scale, 
those  points  which  lie  without  it. 

201.  Niimbers  as  Operators.  The  extension  of  the  number 
system  mentioned  in  the  last  section  may  be  facilitated  by  changing 
the  conception  usually  associated  with  symbols  of  number.  The 
usual  distinction  in  algebra  is  between  symbols  of  number  and  sym- 
bols of  operation.  Thus  a  symbol  which  may  be  looked  upon  as 
answering  the  question  ''bow  many"  is  called  a  number,  while  a 
symbol  which  tells  us  to  do  something  is  called  a  symbol  of  opera- 
tion, or,  simply,  an  operator.  Thus  in  the  expression  \/2,  s/  is 
a  symbol  of  operation  and  2  is  a  number.  A  symbol  of  operation 
may  always  be  read  as  a  verb  in  the  imperative  mood;  thus  we 
may  read  \/x:  "Take  the  square  root  of  x."  Likewise  log  x, 
and  cos  6  may  be  read:  ''Find  the  logarithm  of  x,"  "Take  the  cosine 
of  6,"  etc.     In  these  expressions  "log"  and  "cos"  are  symbols  of 


§201]  COMPLEX  NUMBERS  345 

operation;  they  tell  us  to  do  something;  they  do  not  answer  the 
question  "how  many"  or  "how  much"  and  hence  are  not  numbers. 
Here  we  speak  of  \/,  log,  cos,  as  operators;  we  speak  of  x  as  the 
operand,  or  that  which  is  operated  upon. 

It  is  interesting  to  note  that  any  number  may  be  regarded  as  a 
symbol  of  operation;  by  doing  so  we  very  greatly  enlarge  some 
original  conceptions.  Thus,  10  may  be  regarded  not  only  as  ten, 
answering  the  question  "how  many,"  but  it  may  quite  as  well  be 
regarded  as  denoting  the  operation  of  taking  unity,  or  any  other 
operand  that  follows  it,  ten  times;  to  express  this  we  may  write 
10-1,  in  which  10  may  be  called  a  tensor  (that  is,  "stretcher"), 
or  a  symbol  of  the  operation  of  stretching  a  unit  until  the  result 
obtained  is  tenfold  the  size  of  the  unit  itself.  In  the  same  way 
the  symbol  2  may  be  looked  upon  as  denoting  the  operation  of 
doubling  unity,  or  the  operand  that  follows  it;  likewise  the  tensor 
3  may  be  looked  upon  as  a  trebler,  4  as  a  quadrupler,  etc. 

With  the  usual  understanding  that  any  symbol  of  operation 
operates  upon  that  which  follows  it,  we  may  write  compound 
operators  like  2-2'3-l.  Here  3  denotes  a  trebler  and  3-1  denotes 
that  the  unit  is  to  be  trebled,  2  denotes  that  this  result  is  to  be 
doubled  and  the  next  2  denotes  that  this  result  is  to  be  doubled. 
Thus  representing  the  unit  by  a  line  running  to  the  right,  we  have 
the  following  representation  of  the  operators: 

The  unit  -^ 
31     -^-^-^ 

2-3-1 > > 

2-2-3-1     > > 


Notice  the  significance  that  should  now  be  assigned  to  an  expo- 
nent attached  to  these  (or  other)  symbols  of  operation.  The 
exponent  means  to  repeat  the  operation  designated  by  the  operator; 
that  is,  the  operation  designated  by  the  base  is  to  be  performed, 
and  performed  again  on  this  result,  and  so  on,  the  number  of  opera- 
tions being  denoted  by  the  exponent.  Thus  10^  means  to  perform 
the  operation  of  repeating  unity  ten  times  (indicated  by  10)  and 
then  to  perform  the  operation  of  repeating  the  result  ten  times, 
that  is,  it  means  10  (10-1).  Also,  10'  means  10[10(10-1)].  Like- 
wise log2  30  means  log(]og  30)  which,  if  the  base  be  10,  equals 


346        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§202 

log  1.4771,  or  finally  0.1694^  An  apparent  exception  occurs  in 
the  case  of  the  trigonometric  functions.  The  expression  cos^  x 
should  mean,  in  this  notation,  cos  (cos  x),  but  because  trigonometry- 
is  historically  so  much  older  than  the  ideas  here  expressed,  the 
expression  cos^  x  came  to  be  used  as  an  abbreviation  for  (cos  x)"^, 
or  (cosx)  X  (cosx). 

To  be  consistent  with  the  notation  of  elementary  mathematics, 
the  expression  \/4,  looked  upon  as  a  symbol  of  operation,  must 
denote  an  operation  which  must  be  performed  twice  in  order 
to  be  equivalent  to  the  operation  of  quadrupling;  that  is, 
such  that  (\/4)^  =  4.  Likewise  ^4  denotes  an  operation  which 
must  be  performed  three  times  in  succession  in  order  to  be 
equivalent  to  quadrupling.  But  we  know  that  the  operation 
denoted  by  2,  if  performed  twice,  is  equivalent  to  quadrupling; 
therefore  \/4  =  2,  etc.  Just  as  4^^,  4',  etc.,  may  be  called  stronger 
tensors  than  a  single  4,  so  \/4j  ^i  niay  be  called  weaker  tensors 
than  the  operator  4. 

202.  Reversor.  The  expression  (  —  1),  looked  upon  as  a 
symbol  of  operation,  is  not  a  tensor,  as  it  leaves  the  size  unchanged 
of  that  upon  which  it  operates.  If  this  operator  be  applied  to 
any  magnitude,  it  will  change  the  sense  in  which  the  magnitude 
is  then  taken  to  exactly  the  opposite  sense.  Thus,  if  6  stands 
for  six  hours  after,  then  (  —  1)(6)  stands  for  six  hours  before 
a  certain  event,  and  (  —  1)  is  the  symbol  of  this  operation  of 
reversing  the  sense  of  the  magnitude.  Also  if  (6)  stands  for  a 
line  running  six  units  to  the  right  of  a  certain  point,  then 
(  —  1)(6)  stands  for  a  line  running  six  units  to  the  left  of  that 
point,  so  that  (  —  1)  is  the  symbol  which  denotes  the  opera- 
tion of  turning  the  straight  line  through  180°.  As  2,  3,  4,  when 
looked  upon  as  symbols  of  operations,  were  called  tensors,  the 
operator  (  —  1)  may  conveniently  be  designated  a  reversor. 

Exercises 

Show  graphically  the  effect  of  the  operations  indicated  in  each  of 
the  following  exercises.  Take  as  the  initial  unit-operand  a  straight 
line  1/2  inch  long  extending  to  the  right  of  the  zero  or  initial  point. 
Explain  each  expression  as  consisting  of  the  operand  unity  and  sym- 


§203] 


COMPLEX  NUMBERS 


347 


bols  of  operation — tensors,  reversors,   etc.,  which  operate  upon  it 
one  after  the  other  in  a  definite  order. 


1.  2-3-1. 

2.  3-3-1. 

3.  -1-31. 

4.  23-1. 
6.  V31. 

6.  (-^2)2-1. 

7.  V9V41. 


8.  (V4)3-(-l)-l. 

9.  (-l)3-22-3-l. 

10.  3-321. 

11.  (-l)2-24-l. 

12.  3-(-l)V2-l. 

13.  (V2)-(-l)i°''-l. 

14.  v^l0-2-(-l)-l. 


16.  A  tensor,  if  permitted  to  operate  seven. times  in  succession, 
will  just  double  the  operand.     Symbolize  this  tensor. 

16.  A  tensor,  if  permitted  to  operate  five  times  in  succession,  will 
quadruple  the  operand.     Symbolize  this  tensor. 

203.  Versors.  The  expression  \/  —  1  cannot  consistently, 
with  the  meaning  already  assigned  to  \/  and  ( —  1),  be  looked  upon 
as  answering  the  question 
"how  many,"  and  therefore 
is  not  a  number  in  that  sense; 
yet  if  we  consider  \/—  1  as 
a  symbol  of  operation,  it  can 
be  given  a  meaning  consistent 
with  the  operators  already 
considered.  For  if  2  is  the 
operator  that  doubles,  and 
\/2  is  the  operator  that  when 
used  twice  doubles,  then  since 
(  —  1)  is  the  operator  that 
reverses,  the  expression 
\/—  1  should  be  an  operator 
which,  when  used  twice,  re- 
verses. So,  as  (  —  1)  may  be  defined  as  the  symbol  wh\ch 
operates  to  turn  a  straight  line  through  an  angle  of  180°,  in  a 
similar  way  we  may  define  the  expression  \/^  1  as  a  symbol 
which  denotes  the  operation  of  turning  a  straight  line  through  an 
angle  of  90°  in  the  positive  direction.  The  restriction  of  positive 
rotation  is  inserted  in  the  definition  merely  for  the  sake  of 
convenience. 


B 

(\iny-a 

O               a           ^. 

c 

a 

J 
D 

Fig. 


138. — The  Integral  Powers  of 
V^l. 


348        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§204 

The  symbols  (  —  1)  and  V—  1  are  not  tensors.  They  do  not 
represent  a  stretching  or  contracting  of  the  operand.  Their  effect 
is  merely  to  turn  the  operand  to  a  new  du-ection;  hence  these 
symbols  may  be  called  versors,  or  ''turners." 

204.  Graphic  Representation  of  V—  1-  In  Fig.  138,  let  a  be 
any  line.  Then  a  operated  upon  by  V—  1  (that  is,  \/—  1  a)  is 
turned  anti-clockwise  through  90°,  which  gives  OB.  Now,  of 
course,  -%/—  1  can  operate  on  V—  1  «  just  as  well  as  on  a. 
Then  \/—  1  [\/—  1  a),  or  OC,  is  \/—  1  «  turned  pcsi- 
tively  through  90°.  Similarly,  \/^T  [\/^^(\/^n[  a)]  is 
V-  1  (V-  1  «)  'turned  through  90°,  etc. 

As  we  are  at  liberty  to  consider  two  turns  of  90°  as  equivalent  to 
one  turn  of  180°,  therefore,  \/—  1(\/—  1  «)  =  (  —  !)«•  Now 
OP  =  (-1)  OB,  0D={-1)  (V^^la);  but  also  OD  = 
V  —  1  (—  «),  therefore,  (  —  1 )  \/—  1  a  =  \/—  1  (  —  a). 
Thus  the  student  may  show  many  like  relations. 

The  operator  \/  —  1  is  usually  represented  by  the  symbol  i  and 
will  generally  be  so  represented  in  what  follows 

Exercises 

Interpret  each  of  the  following  expressions  as  a  symbol  of  operation: 

1.  2,3,4,  -1. 

2.  32,   23,40,    (-1)2,    (-1)6. 

3.  V2,  V3,  V^l,  V%  V^^' 

Select  a  convenient  unit  and  construct  each  of  the  following  expres- 
sions geometrically,  explaining  the  meaning  of  each  operator: 

4.  2-3-5-1.  7.  (-1)2V^^-1. 

5.  23-(-l)-l.  8.  22-(-l)3-(V-  i)°-l. 

6.  3V^n-21.  9.  3V^=n[-(-l)V^^-l. 

205.  Complex  Numbers.  The  explanation  of  the  meaning  of 
the  symbol  (a  +  bi)  will  be  given  in  the  following  section.  It 
will  be  shown  in  subsequent  theorems  that  any  expression  made 
up  of  the  sum,  product,  power  or  quotient  of  real  numbers  and 
imaginaries  may  be  put  in  the  form  a  +  bi,  in  which  both  a  and 
b  are  real.  The  expression  a  -\-  bi  is  therefore  said  to  be  the 
typical  form  of  the  imaginary.     An  expression  of  the  form  a  +  bi 


§206] 


COMPLEX  NUMBERS 


349 


is  also  called  a  complex  number,  since  it  contains  a  term  taken 
from  each  of  the  following  scales,  so  that  the  unit  is  not  single 
but  double  or  complex: 

•    .    •    -  3,    -  2,    -  1,  0,  +  1,  +  2,    +  3,    .    .    . 
•■'-Si,-  2i,  -  i,  0,  +  i,  +  2i,  +  Si,   .    .    . 

It  is  important  to  note  that  the  only  element  common  to  the  two 
series  in  this  complex  scale  is  0. 

206.  Meaning  of  a  Complex  Nimiber.  Any  real  number,  or 
any  expression  containing  only  real  numbers,  may  be  consid- 
ered as  locating  a  point  in  a  line. 

Thus,  suppose  we  wish  to  draw  the  expression  2+5.  Let  0  be 
the  zero  point  and  OX  the  positive  direction.  Lay  off  OA  =  2  in 
the  direction  OX  and  at  A  lay  ofi  AB  =  5  in  the  direction  OX. 
Then  the  path  OA  +  AB  is  the  geometrical  representation  of 
2+5. 

0  A  B  X 


h.P 


Any  complex  number  may  be  taken  as  the  representation  of  the 
position  of  a  point  in  a  plane.  For,  suppose  c  +  di  is  the  complex 
number.  Let  0  be  the  zero 
point  and  OX  the  positive 
direction.  Lay  off  OA  =  +  c 
in  the  direction  OX  and  at 
A  erect  di  in  the  direction 
OY,  instead  of  in  the  direction 
OX  as  in  the  last  example. 
See  Fig.  139.  It  is  agreed 
to  consider  the  step  to  the 
right,  OA,  followed  by  the 
step  upward,  AP,  as  the 
meaning  of  the  complex  num- 
ber c  +  di.  Either  the  broken  path  OA  +  AP  or  the  direct  'path 
OP  may  he  taken  as  the  representation  of  c  -{-  di,  and  either  path 
constitutes  the  definition  of  the  sum  of  c  and  di. 

In  the  same  manner  c  —  di,  —  c  —  di  and  —  c  -{-  di  may  be 
constructed. 

The  meaning  of  some  of  the  laws  of  algebra  as  appUed  to  imagi- 
naries  may  now  be  illustrated.     Let  us  construct  c  +  di  +  a  +  bi. 


Fig.  139.     The  Geometrical  Construc- 
tion of  a  Complex  Number,  c  +  di. 


350        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§207 


The  first  two  terms,  c  +  di,  give  OA  -{-  AB,  locating  B  (Fig. 
140).  The  next  two  terms,  a  +  bi,  give  BC  +  CP,  locating  P. 
Hence  the  entire  expression  locates  the  point  P  with  reference  to  0. 
Now  if  the  original  expression  be  changed  in  any  manner  allowed  by 
the  laws  of  algebra,  the  result  is  merely  a  different  path  to  the  same 
point.     Thus: 

c  +  a   +di  + hi  is  the  path  OA,  AD,  DC,  CP 
(c  +  a)  +  {d  +  b)i  is  the  path  OD,  DP 
a  +  di+c    +bi   is  the  path  OE,  EH,  HC,  CP 
a-\-di  +  M  +c     is  the  path  OE,  EH,  HF,  FP,  etc. 
The  student  should  consider  other  cases.     Are  there  any  method 
of  locating  P  with  the  same  four  elements,  which  the  figure  does  not 
illustrate? 


Y 

a 

c 

—  ^ 

-or- 

1 
1 

•£ 

JP 

•t» 

1 

'—^ 

1 

1 

+ 

I 

«! 

b\ 

a        C 

-a 

1 

••si 

1 

-5 

, 

': 

V.P 

E\ 

Ix 

V 

A 

l> 

Fig. 


140.  Illustration  of  the  Application  of  the  Laws  of  Algebra  to  the 
Expression  c  -\-  di  -\-  a  -{-  bi. 


207.  Laws.  It  can  be  shown  by  simple  geometrical  construc- 
tion that  the  operator  {,  as  defined  above,  obeys  the  ordinary 
laws  of  algebra.  We  can  then  apply  all  of  the  elementary  laws  of 
alegbra  to  the  symbol  i  and  work  with  it  just  as  we  do  with  any 
other  letter.     The  following  are  illustrations  of  each  law: 

Commutative  Law: 

c-\-di-{-a  +  hi  —  c-j-a-\-di-\-bi  =  di  +  c  +  bi-\-a,  etc. 
ai  =  ia,  iai  =  iia  =  aii,  etc. 

The  equation  10\/^  =  V^'IO,  or  better,  10\/^-l  = 
\/  —  I'lO-l  may  be  said  to  mean  that  the  result  of  performing  the 
operation  of  turning  unity  through  90°  and  performing  upon  the 


§208]  COMPLEX  NUMBERS  351 

result  the  operation  of  taking  it  ten  times,  is  the  same  as  the  result 
of  performing  the  operation  of  taking  unity  ten  times  and  per- 
frrming  upon  this  result  the  operation  of  turning  through  90°. 

Associative  Law: 

(c  +  di)  +  (a  +  bi)    =  c  +  {di  -\-  a)  +  hi,  etc. 
{ab)i  =  a{bi)  =  ahi,  etc. 

Distributive  Law: 

(a  +  h)i  =  ai  +  hi,  etc. 


The  expression  \/—  a,  where  a  is  any  number  of  the  arith- 
metical scale,  is  defined  as  equivalent  to  \/  — l-a;that  is,  V—  a 
=  i\/a.  For  example,  \/—  4  =  2i,  \/—  3  =  i\/Z,  etc.  In 
what  follows  it  is  presupposed  that  the  student  will  reduce  expressions 
of  the  form  \/  —  a  to  the  form  i  \/a  hefore  performing  algebraic  op- 
erations. From  this  it  follows  that  \/  —  a  \/'  —  b  =  —  \/ab  and 
not  \/ab. 

The  relation  v"^-^  =  2  \/  — 1  i^ay  be  interpreted  as  follows:  ( —  4) 
is  the  operator  that  quadruples  and  reverses;  then  V  —  4  is  an  operator 
which  used  twice  quadruples  and  reverses.  But  2  \/  —  1  is  an  opera- 
tor such  that  two  such  operators  quadruple  and  reverse.  That  is, 
\/^^4  =  2^/^. 

208.  Powers  of  i.  We  shall  now  interpret  the  powers  of  i  by 
means  of  the  new  significance  of  an  exponent  and  by  the  commu- 
tative, associative  and  other  laws.     First: 

1°  or  i^'  1  =  +  1  ^^  =  i^  =       i 

i^  or  i^  1  =       i  i^  =  iH  =  —  1 

i2  =  —  1  iT  =  iH  =  —  i 

i^  =  iH  =  —  i  i^  =  iH  =  +  1 

i*  =  iH^  =  +  1                     etc.  etc. 

Whence  it  is  seen  that  all  even  powers  of  i  are  either  -f  1  or  —  1, 
and  all  odd  powers  are  either  i  or  —  i.  The  student  may  reconcile 
this  with  Fig.  138.  The  zero  power  of  i  must  be  unity,  for  the 
exponent  zero  can  only  mean  that  the  operation  denoted  by  the 
symbol  of  operation  is  not  to  be  performed  at  all;  that  is,  unity  is  to 
be  left  unchanged;  thus  10°  or  10°  •  1  =  1,  2°  =  1,  log^  x  =  x, 
sin"  X  =  X,  etc. 


352        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§210 

Exercises 

Select  as  unit  a  distance  1/2  inch  in  length  extending  to  the  right 
and  represent  graphically  each  of  the  following  expressions: 

1.  i  +  2i2  +  3i3  +  4^4  +    .    .    . 

2.  i-\-i*  +  i^  +  i^  +  i^  +   .    .    . 

3.  i  +  i*  +  ^7  +  i8  +  ^9  _|_  ji2  _^   ,    ^    , 

4.  i(i  +  t*  +  i7  +  i«  +  i9+ii2+   .    .    .   ). 

5.  i  +  i2  +  i3  _{_  2^4  4-  i5  -I-  i6  _j_  ^7  +  3j;s  +    .    ,    . 

209.  Two  complex  numbers  are  said  to  be  conjugate  if  they 
differ  only  in  the  sign  of  the  term  containing  \/—  1.  Such  are 
X  +  iy  and  x  —  iy. 

Conjugate  imaginaries  have  a  real  sum  and  a  real  product. 

For:  {x  +  yi)  +  (^  -  yi) 

—  X  -\-  yi  -'t-  X  —  yi,  by  associative  law 
=  X  -\-  X  -\-  yi  —  yi,  hy  commutative  law 
=  2x  -{-  {yi  —  yi),  by  associative  law 
=  2x  -\-  {y  —  y)i,  by  distributive  law 

Likewise:   {x  +  yi){x  —  yi) 

=  x{x  —  yi)  +  yi{x  —  yi),  by  distributive  law 
=  x^  —  xyi  +  yix  —  yiyi,  by  distributive  law 
=  x"^  —  yH"^  +  xyi  —  xyi,  by  commutative  law 
=  x^  +  2/2  +  {xy  —  xy)i,  by  distributive  law  and  by 

substituting  i"^  =  —  1 

=    X2    +  2/2 

It  is  well  to  note  that  the  product  of  two  conjugate  complex 
numbers  is  always  positive  and  is  the  sum  of  two  squares. 

This  fact  is  very  important  and  will  be  frequently  used.     Thus 
(3  -  4i)(3  +  4i)  =  32  +  42  =  25;  (1  +  i)(l  -  i)  =  2 
(cos  0  +  i  sin  6){cos  d  —  i  sin  d)  =  cos2  6  +  sin2  ^  =  1,  etc. 

210.  The  sum,  product,  or  quotient  of  two  complex  numbers  is, 
in  general,  a  complex  number  of  the  typical  form  a  +  bi. 

Let  the  two  complex  numbers  be  a;  +  yi  and  u  +  vi. 
(1)  Their  sum  is  {x  +  yi)  -\-  (u  +  vi) 

=  X  -\-  yi  -\-  u  -\-  vi 

=  X  -{-  u  -\-  yi  -{-  vi 

=  {x  -\-  u)  +  {y  +  v)i 


§210]  COMPLEX  NUMBERS  353 

by  the  laws  of  algebra.     This  last  expression*  is  in  the  form  a  +  bi. 

(2)  Their  product  is  {x  +  yi)  (u  +  vi) 

=  x{u  -\-  vi)  +  yi{u  +  vi) 
=  zu  -\-  xvi  +  yiu  +  yivi 
=  xu  -{-  yvi^  +  xvi  +  yui 
=  {xu  —  yv)  +  (xv  +  yu)i 
by  the  laws  of  algebra.     This  last  expression  is  in  the  form  a  +  bi. 

(3)  Their  quotient  is 

X  +  yi  ^  (x  +  j/t)(^  -  vi) 

u  -\-  vi  ~  (u  -{-  vi){u  —  vi) 

By  the  preceding,  the  numerator  is  of  the  form  a'  +  b'i.     By 

§209,  the  denominator  equals  u^  +  y^.     Then  the  quotient  equals 

a'  +  b'i  a'  b'       . 


^2  _|_  y2  ^2  _j_  y2      -      ^2  _^  ^2   - 

by  distributive  law.     This  last  expression  is  of  the  form  a  +  bi. 

Exercises 

Reduce  the  following  expressions  to  the  typical  forin  a  -\-  bi; 
the  student  must  change  every  imaginary  of  the  form  V  -  o  to  the 
form  i  Va. 

1.  V-25  +  V^^  +  V^121  -  V  -  64  -  6i 

2.  (2V^  +  3V^)(4V^  -SV"^). 

3.  {x  -  [2.+  3i])(x  -  [2  -3-6]). 

4.  (-  5  +  12V -1)2.  6.  (Vl  +  i)(Vl  -i). 
6.  (3-4V^)2.                                7.  (N/7-V-'e)2. 

1 


8. 

a 

V-1 

2 

9. 

3  +  V- 

2 

10. 

56 

1-V^ 

-7 

11 

1+* 

12. 
13. 
14. 
15. 


(1  -  i)' 
1  -i» 

a-i)''  _ 
1  -  2V  -  3 

1  +  2V^3' 

(2+3V'^T)« 


1  -  i«  2  +  V 

a  -\-  oci       a  —  xi 


16.  .  j_    . 


23 


354        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§211 

211.  Irrational  Numbers.  A  rational  number  is  a  number  that 
can  be  expressed  as  the  quotient  of  two  integers.  All  other  real 
numbers  are  irrational.  Thus  \/2,  \/5,  \/'7,Tr,  e,  are  irrational 
numbers.  An  irrational  number  is  always  intermediate  in  value 
to  two  rational  numbers  which  differ  from  each  other  by  a  number 
as  small  as  we  please.     Thus 

1.414      <  \/2  <  1.415 
1.4142     <  V2  <  1.4143 
1.41421  <  \/'2  <  1.41422,  etc. 

It  is  easy  to  prove  that  \/2  cannot  be  expressed  as  the  quotient 
of  two  integers.     For,  if  possible,  let 

V2  =  I  (1) 

where  a  and  h  are  integers  and  t  is  in  its  lowest  terms.     Squaring 
the  members  of  (1)  we  have 


a2 
2  =  (^  (2) 

This  cannot  be  true,  since  2  is  an  integer  and  a  and  h  are  prime 
to  each  other. 

An  irrational  number,  when  expressed  in  the  decimal  scale,  is 
never  a  repeating  decimal.  For  if  the  irrational  number  could  be 
expressed  in  that  manner,  the  repeating  decimal  could  be  evalu- 
ated by  §123  in  the  fractional  form  y-^ — >  which,  by  definition 

of  an  irrational  number,  is  impossible.  On  the  contrary,  every 
rational  number  when  expressed  in  the  decimal  scale  is  a  repeating 
decimal.     Thus  1/3  =  0.33   ...  and  1/4  =  0.25000.   .    .    . 

The  proof  that  tt  and  e  are  irrational  numbers  is  not  given  in 
this  book. 

See  Monographs  on  Modern  Mathematics,  edited  by  J.  W.  A. 
Young. 

The  student  should  not  get  the  idea  that  because  irrat'onal  numbers 
are  usually  approximated  by  decimal  fractions,  that  the  irrational 
number  itself  is  not  exact.  This  can  be  illustrated  by  the  graphical 
construction  of  \/2-  Locate  the  point  P  whose  coordinates  are  (1,1). 
Call  the  abscissa  OD  and  the  ordinate  DP.  Then  OP  =  \/2  and 
OD  =  1,  DP  =  1.     It  is  obvious  that  the  hypotenuse  OP  must  be 


§212]     .  COMPLEX  NUMBERS  355 

considered  just  as  exact  or  definite  as  the  legs  OD  and  DP.     The 
notion  that  irrational  numbers  are  inexact  must  be  avoided. 

The  process  of  counting  objects  can  be  carried  out  by  use  of  the 
primitive  scale  of  numbers  0,  1,  2,  3,  4,  .  .  .  The  other  numbers 
made  use  of  in  mathematics,  namely, 

(1)  positive  and  negative  numbers 

(2)  integral  and  fractional  numbers 

(3)  rational  and  irrational  numbers 

(4)  real  and  imaginary  numbers 

may  be  looked  upon  as  classes  of  numbers  that  permit  the  opera- 
tions subtraction,  division  and  evolution,  to  be  carried  out  under  all 
circumstances.  Thus,  in  the  history  of  algebra  it  was  found  that  in 
order  to  carry  out  subtraction  under  all  circumstances,  negative  num- 
bers were  required;  to  carry  out  division  under  all  circumstances,  frac 
tions  were  required;  to  carry  out  evolution  of  arithmetical  numbers 
under  all  circumstances,  irrational  numbers  were  required;  finally  to 
carry  out  evolution  of  algebraic  numbers  under  all  circumstances, 
imaginaries  were  required.  It  will  be  found  that  it  will  not  be  neces- 
sary to  introduce  any  additional  form  of  number  into  algebra;  that 
is,  the  most  general  number  required  is  a  number  of  the  form  a  +  6i, 
where  a  and  b  are  positive  or  negative,  integral  or  fractional,  rational 
or  irrational.  This  is  the  most  general  number  that  satisfies  the  fol- 
lowing conditions: 

(a)  The  possibiUty  of  performing  the  operations  of  algebra  and 
the  inverse  operations  under  all  circumstances. 

(6)  The  conservation  or  permanence  of  the  fundamental  laws  of 
algebra:  namely,  the  commutative,  associative,  distributive  and  index 
laws. 

Further  extension  of  the  number  system  beyond  that  of  complex 
numbers  leads  to  operators  which  do  not  obey  the  commutative  law  in 
multiplication;  that  is,  in  which  the  value  of  a  product  is  dependent 
upon  the  order  of  the  factors,  and  in  which  a  product  does  not  neces- 
sarily vanish  when  one  factor  is  zero.  Numbers  of  this  kind  the 
student  may  later  study  in  the  introduction  to  the  study  of  electro- 
magnetic theory  under  the  head  of  "Vector  Analysis"  or  in  the 
subject  of  "Quaternions."  Such  numbers  or  operators  do  not  belong 
to  the  domain  of  numbers  we  are  now  studying. 

212.  If  a  complex  number  is  equal  to  zero,  the  imaginary  and 
real  parts  are  separately  equal  to  zero. 
Suppose  X  -{-  y  y  —1  =  0 

then  X  =  —  y  yl  —  1 


356        ELEMENTARY  MATHEMATICAL  ANALYSIS.     [§214 

Now  it  is  absurd  or  impossible  that  a  real  number  should  equal 
an  imaginary,  except  they  each  be  zero,  since  the  real  and  imagi- 
nary scales  are  at  right  angles  to  each  other  and  intersect  only  at 
the  point  zero. 
Therefore:  x  =  0  and  y  =  0 

If  two  complex  numbers  are  equal,  then  the  real  and  the  imaginary 
parts  must  be  respectively  equal. 

For  if  X  +  yi  =  u  -]-  vi 

then  (x  —  u)  -{-  {y  —  v)i  =  0 

Whence,  by  the  above  theorem, 

X  —  u  =  0  and  y  —  v  =  0 

That  iS;  X  =  u  and  y  =  v 

213.  Modulus.  Let  the  complex  number  x  -]-  yihe  constructed, 
as  in  Fig.  141,  in  which  OA  =  x  and  AP  =  yi.  Draw  the  Jine 
OP,  and  let  the  angle  AOP  be  called  d. 

The  numerical  length  of  OP  is  called  the  modulus  of  the  complex 
number  x  +  yi.  It  is  algebraically  represented  by  Va;^  _^  yz^ 
or  by  \x  +  yi\.     Thus,  mod  (3  +  4i)  =  V9  +  16  =  5. 

The  student  can  easily  see 
that  tivo  conjugate  complex 
numbers  have  the  same  modu- 
lus, which  is  the  positive  value 
of  the  square  root  of  their 
product. 

If  y  =  0,  the  mod  {x  -\-  yi) 

=  Vx^    =     \x\,   where    the 
Fig.  141. — Modulus   and  Amplitude  of   ,,^„j.-  „i   i-   ^„    ,-^ri,"««+^  +!.«  + 
a  Complex  Number.  ^^^^^^^^   ^'^^^    mdicate  that 

merely  the  numerical  or 
absolute  value  of  x  is  called  for.  Thus  the  modulus  of  any  real 
number  is  the  same  as  what  is  called  the  numerical  or  absolute 
value  of  the  number.     Thus,  mod  (—5)  =5. 

214.  Amplitude.  In  Fig.  141  the  angle  AOP  or  d  is  called  the 
argument  or  amplitude  or  simply  the  angle  of  the  complex  number 
X  +  yi. 


j215]  COMPLEX  NUMBERS  357 


Putting  r  =  '^x^  -\-  y^  =  mod  {x  +  yi),  we  have 

y  X 

sin  6  =  —  cos  ^  =  - 

r  r 

Therefore, 

X  -\-  yi  =  r  cos  d  +  tV  sin  6  =  r(cos  ^  +  i  sin  6) 

in  which  we  have  expressed  the  complex  number  x  +  yi  in  terms  of 

its  modulus  and  amplitude. 

To  put  3  —  4i  in  this  form,  we  have: 

mod  (3  -  4i)  =  Vq  +  16  =  5;    sin  0  =  —  =  -  ^:    cos  9  =  —  =  ^ 

r  o  TO 

Therefore, 

(3-4i)=5[|-|i] 

The  amphtude  ^  is  tan~^  (  ~  ^ )  >  ^^^  ^  ^^  ihe fourth  quadrant.    Why  ? 

It  is  well  to  plot  the  complex  number  in  order  to  be  sure  of  the  ampli- 
tude 6.  It  avoids  confusion  to  use  positive  angles  in  all  cases.  For 
example,  to  change  3  —  V  3  i  to  the  polar  form,  plot  the  point 
(3,  -  V  3)  and  find  from  the  triangle  that  r  =  2  V  3  and  0  =  330^ 
Hence 

3  _  V  3  t  =  2  V  3(cos  330°  +  i  sin  330°) 

The  amplitude  of  all  positive  numbers  is  0,  and  of  all  negative 
numbers  is  180°.  The  unit  expressed  in  terms  of  its  modulus  and 
amplitude  is  evidently  l(cos  0  +  i  sin  0). 

215.  Vector.  The  point  P,  located  by  OA  +  AP  or  a;  +  yi, 
may  also  be  considered  as  located  by  the  line  or  radius  vector  OP; 
that  is,  by  a  line  starting  at  0,  of  length  r  and  making  an  angle  d 
with  the  direction  OX.  A  directed  line,  as  we  are  now  considering 
OP,  is  called  a  vector.  When  thus  considered,  the  two  parts  of  the 
compound  operator 

r  (cos  d  -\-  i  sin  6)  (1) 

receive  the  following  interpretation :  The  operator  (cos  ^  +  i  sin  6) , 
which  depends  upon  d  alone,  turns  the  unit  lying  along  OX 
through  an  angle  6,  and  may  therefore  be  looked  upon  as  a 
versor  of  rotative  power  0.  The  versor  (cos  d  -{■  i  sin  d)  is  often 
abbreviated  by  the  convenient  symbol  cis  6.  The  operator  r 
is  a  tensor,  which  stretches  the  turned  unit  in  the  ratio  r  :  1.  The 
result  of  these  two  operations  is  that  the  point  P  is  located  r  units 
from  0  in  a  direction  making  the  angle  0  with  OX. 


358        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§216 

The  operator  (1)  above  is  also  represented  by  the  notation 
(r,  6),  for  example  (5,  /30°).  Expression  (1)  is  called  the  polar 
form  of  the  complex  number  (a;+  iy). 

Thus,  the  operator  (cos  6  -{-  i  sin  6)  is  simply  a  more  general 
operator  than  i,  but  of  the  same  kind.  The  operator  i  turns 
a  unit  through  a  right  angle  and  the  operator  (cos  d  -\-  i  sin  6) 
turns  a  unit  through  an  angle  d.  If  6  be  put  equal  to  90°, 
cos  6  -^  i  sin  6  reduces  to  i. 

For:  ^  =  0,  cos  ^  +  ^  sin  6  reduces  to  1 
d  =  90°,  cos  d  -\-  i  smd  reduces  to  i 
6  =  180°,  cos  6  -\-  i  sin  6  reduces  to  —  1 
6  =  270°,  cos  ^  +  ?;  sin  ^  reduces  to  —  i 

Since  3  —  4i  =  5(f  —  fi)  the  point  located  by  3  —  4i  may  be 
reached  by  turning  the  unit  vector  through  an  angle  6  = 
sin~^  (—4/5)  =  cos-i  3/5  and  stretching  the  result  in  the 
ratio  5:1. 

If  a  complex  number  vanishes,  its  modulus  vanishes;  and  con- 
versely, if  the  modulus  vanishes,  the  complex  number  vanishes. 

If  a;  +  yi  =  0,  then  a;  =  0   and   y  =  0,  by  §212.     Therefore, 

Vx2  +  2/2  =  0.  Also,  if  yjx^  -\- y^  =  0,  then  x^ -\- y"^  =  0,  and 
since  x  and  y  are  real,  neither  x^  nor  i/^  is  negative,  and  so  their 
sum  is  not  zero  unless  each  be  zero. 

If  two  complex  numbers  are  equal,  their  moduli  are  equal,  but  if 
two  moduli  are  equal,  the  complex  numbers  are  not  necessarily  equal. 

li x  -{-  yi  =  u  -{■  vi,  then x  =  u  and  y  =  vhy  §212. 

Therefore,      '^x^  -h  y^  =  ^u^  +  v^ 

But  if  Vx^  -f  2/^  =  "Vti^  +  1^2^  obviously  x"^  need  not  equal 

u"^  nor  2/2  =  y2. 

216.  Sum  of  Complex  Numbers.  Let  a  given  complex  number 
locate  the  point  A,  Fig.  142,  and  let  a  second  complex  number 
locate  the  point  B.  Then  if  the  first  of  the  complex  numbers  be 
represented  by  the  radius  vector  OA  and  if  the  second  complex 
number  be  represented  by  the  radius  vector  OB,  the  sum  of  the  two 
complex  numbers  will  be  represented  by  the  diagonal  OC  of  the 
parallelogram  constructed  on  the  lines  OA  and  OB.  This  law  of 
addition  is  the  well-known  law  of  addition  of  vectors  used  in  physics 
when  the  resultant  of  two  forces  or  the  resultant  of  two  velocities, 


/ 


§216] 


COMPLEX  NUMBERS 


359 


two  accelerations,  or  two  directed  magnitudes  of  any  kind,  is  to  be 
found. 

The  proof  that  the  sum  of  the  two  complex  numbers  is  repre- 
sented by  the  diagonal  OC  is  very  simple.  Let  the  graph  of  the  first 
complex  number  be  ODi  -\-  DiA  and  let  that  of  the  second  be  OD2 
+  D2B.  To  add  these,  at  the  point  A  construct  AE  =  OD2  and 
EC  =  D2B.  Then  the  sum  of  the  two  complex  numbers  is  geo- 
metrically represented  by  ODi  +  DiA  -\-  AE  +  EC,  or  by  the 
radius  vector  OC  which  joins  the  end  points.  Since,  by  construc- 
tion, the  triangle  AEC  is  equal  to  the  triangle  02)25,  therefore  AC 


Fig.  142. — Sum  of  Two  Complex  Numbers. 

must  be  equal  and  parallel  to  OB,  so  that  the  figure  OACB  is  a 
parallelogram,  and  OC,  which  represents  the  required  sum,  is  the 
diagonal  of  this  parallelogram,  which  we  were  required  to  prove. 


Exercises 

Find  algebraically  the  sum  of  the  following  complex  numbers,  and 
construct  the  same  by  means  of  the  law  of  addition  of  vectors. 

1.  (1  +  2i)  +  (3  +  4i). 

2.  (1  +  i)  +  (2  +  i). 

3.  (1  -i)  +  (1  +2i). 

4.  (3  -  4i)  -  (3  -h  4i). 
6.  (-  2  +i)  +  (0  -  4i). 

Q.  (-  I  +i)  +{S+i)  +(2+  2i). 


360        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§217 

7.  (2  -i)  +  (-2  +^)  +  (1  +i).. 

8.  Find  the  modulus  and  amplitude  (in  degrees  and  minutes)  of 
2  (cos  30°  +  i  sin  30°)  +  (cos  45°  +  i  sin  45°). 

9.  By  the  parallelogram  of  vectors,  show  that  the  sum  of  two  con- 
jugate complex  numbers  is  real. 

10.  If  R  be  the  sum  of  the  complex  numbers  Zi  =  Xi  -\-  iyi,  Zi  = 
X2  +  iyi,  zz  =  xz  -\-  yzi,  etc.,  show  that  —  R,  Z\,  Zi,  Zz,  .  .  .  form  the 
sides  of  a  closed  polygon. 

217.  Polar  Diagrams  of  Periodic  Functions.  Three  methods  of 
representing  simple  periodic  phenomena  have  already  been  ex- 
plained: (1)  Crank  or  clock  diagrams  as  shown  in  Fig.  128  and 
explained  in  §184;  (2)  the  sine  curve  or  sinusoid  in  rectangular 
coordinates,  as  shown  in  Fig.  59  and  explained  in  §55; 
(3)  polar  diagrams,  in  which  the  circle  (twice  drawn)  corresponds 
to  a  crest  and  trough  of  the  sine  curve,  as  shown  in  Fig.  63  and 
explained  in  §64.  As  the  principal  application  of  these  methods 
is  to  phenomena  that  vary  with  the  time,  one  of  the  variables 
may  conveniently  be  taken  to  represent  time  or  a  constant  multi- 
ple of  time.  Thus  the  angle  6  in  the  crank  and  polar  diagrams  or 
the  abscissa  in  the  Cartesian  diagram,  may  be  represented  by 
a  constant  multiple  of  t  as  cot. 

The  difference  between  a  clock  diagram  and  a  polar  diagram 
of  a  simple  periodic  function  may  be  stated  as  follows:  In  a 
clock  diagram,  a  rotating  vector  of  fixed  length  OP  is  continu- 
ously projected  upon  a  fixed  line  OX;  in  a  polar  diagram,  a 
stationary  line  of  fixed  length  OA  is  continuously  projected 
upon  a  rotating  radius  vector  OP.    See  Figs.  52  and  63. 

Each  of  the  three  methods  possesses  a  peculiar  advantage  of  its 
own,  but  probably  the  best  insight  with  regard  to  periodic  phe- 
nomena is  given  by  the  polar  diagram.  In  each,  the  complete 
period  of  the  phenomena  is  represented  by  one  complete  revolution 
of  the  radius  vector.  The  polar  method  is  not  only  well  adapted 
to  represent  simply  varying  periodic  phenomena,  in  which  case  the 
polar  diagram  is  a  circle  passing  through  the  origin,  but  it  is 
equally  well  adapted  to  represent  cases  in  which  the  periodic 
motion  is  compounded  from  a  number  of  simple  harmonics.  In 
many  important  cases  in  science,  especially  in  the  phenomena  of 
alternating  electric  currents,  only  the  odd  harmonics  are  commonly 
present  as  components  of  the  resulting  motion.     The  equation  of 


§217]  COMPLEX  NUMBERS  361 

compound  harmonic  motion  in  rectangular  coordinates,  in  which 
only  odd  harmonics  appear,  is  of  the  form: 

y  =  tti  sin  cot  +  az  sin  3cx)t  +  a^  sin  5cot  +   •    •    •  (1) 

in  which  Oi,  az  .  .  .  are  any  constants  and  in  which  o)t  has  re- 
placed x.  If  the  epochs^  of  the  various  harmonics  be  <i,  <3,  ^5,  •  •  • 
the  proper  form  of  the  equation  would  be: 

y  =  ai  sin  uj{t  —  ti)  +  clz  sin  So}{t  —  ^3)  +  a^  sin  5co(i  —  ^5) 

+   .    .    .  (2) 

A  curve  of  type  (1)  or  (2)  must  represent  a  pattern  within  the  inter- 
val oit  =  IT  to  cot  =  2ir  which  is  the  opposite  of  the  pattern  pre- 
sented within  the  interval  oot  =  0  io  oot  =  t]  for  increasing  cot 
by  the  amount  tt  in  each  of  the  components: 

sin  cot,  sin  3co^,  sin  5  coi  .  .  . 
of  the  compound  motion  has  the  effect  of  changing  the  algebraic 
sign  of  each  term,  but  leaves  the  absolute  value  unchanged.  This 
is  because  the  sine  curve,  and  all  of  the  odd  harmonics  of  the  sine 
curve,  are  just  reversed  in  sign  by  adding  a  straight  angle  (180°) 
or  an  odd  number  of  straight  angles  to  the  original  angle.  Hence 
y  has  the  same  sequence  of  values,  but  of  opposite  signs,  within 
each  of  the  two  half-intervals  of  the  period  27r.  Fig.  143  illustrates 
this.  The  curve  A  is  the  graph  of  an  alternating  current  wave 
(after  Fleming)  in  rectangular  coordinates,  while  the  same  func- 
tion is  shown  in  polar  coordinates  by  curve  B.  It  is  observed  that 
the  second  portion  of  the  Cartesian  curve  is  exactly  similar  to  the 
first  portion,  except  that  its  position  with  reference  to  the  x-axis 
is  reversed.  In  the  polar  diagram  this  truth  is  brought  out  by 
the  fact  that  the  loop  that  represents  the  ''trough"  of  the  wave 
is  identical  with  the  loop  that  represents  the  "crest"  of  the  wave, 
that  is,  the  curve  is  twice  drawn  to  represent  the  interval  of  a 
complete  period  from  co^  =  0  to  coi  =  2ir. 

If  only  even  harmonics  are  present,  the  equation  of  the  curve  in 
rectangular  coordinates  is  of  the  form: 

y  =  aQ-\-  at  sin  2o3t  -f-  «4  sin  4coi  +  .    •    •  (3) 

or,  if  the  epochs  are  not  zero, 

2/  =  ao  +  02  sin  2aj(^  —  ^2)  +  ^4  sin  4co(i  —  ^4)  +  .    •    .     (4) 

1  This  expression  insteed  of  "epoch  angle"  is  the  proper  term  in  this  case  as  i  is 
measured  in  units  of  time  and  not  in  angular  measure.  The  epoch  angles  are  uh, 
<ati,  etc. 


362        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§217 

Because  of  the  factor  2  in  each  harmonic  term,  the  period  of  the 
function  may  be  considered  tt  instead  of  27r,  so  that  the  sequence  of 
values  of  y  are  repeated  in  each  interval  0  to  tt,  tt  to  2ir,  etc.,  and 
not  reversed  in  sign  as  in  the  case  of  the  odd  harmonics. 

The  fact  that  the  pattern  for  each  interval  x  is  repeated  right 
side  up  and  not  reversed  is  illustrated  by  the  graph  of 

2/  =  1  +  sin  2oit  4-  sin  4aj^  (5) 


Fig.  143. — Rectangular    and  Polar   Alternating  Current  Curves.      {After 

Fleming.) 

shown  in  Fig.  144.     The  effect  of  the  constant  term  is,  of  course? 
merely  to  raise  the  graph  a  distance  of  one  unit. 

In  jorm,  curves  with  only  even  harmonics  present  do  not  differ 
from  curves  with  both  odd  and  even,  for  substituting  t'  =  2t,  the 
general  case  (equation  (3))  becomes: 

?/  =  ao  +  a2  sin  oit'  +  ^4  sin  2cof  +  «6  sin  3coi'  +  .    .    .     (6) 

which  contains  both  odd  and  even  harmonics  in  t',  and  is  of  period 
27r.     The  curve  (3)  is  of  the  same  shape  as  (6)  but  of  period  tt. 


§217] 


COMPLEX  NUMBERS 


363 


A  curve  made  up  of  both  odd  and  even  harmonics  may  have  any 
form  consistent  with  its  one-valued  and  continuous  character.  The 
portion  of  the  curve  above  the  a;-axis  (if  any)  need  not  have  the 
same.form  as  the  part  below;  the  only  essential  is  that  the  curve  for 


Fig.  144. — Graph  oi  y  =  sin  2  wt  -\-  sin  4  cot. 

each  successive  interval  of  27r  be  a  repetition  of  the  curve  in  the 
preceding  interval. 

In  polar  coordinates,  a  curve  made  up  of  only  even  harmonics 
is  described  but  once  as  d  varies  from  0°  to  360°.     In  general  such 


Fig.  145.— Graph  of  p  =  sin  2e  +  sin  4e. 

curves  have  more  "loops"  than  curves  made  up  of  odd  harmonics, 
for  the  loops  of  the  odd  harmonics  are  twice  drawn  as  d  varies 
from  0°  to  360°.     Thus  the  curves: 

p  =  sin  6,  p  =  sin  3^,  p  =  sin  5^,   .    .    . 


364        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§217 

have  1,  3,  5,   .    .    .  loops  respectively  each  twice  drawn.    The 
curves 

p  =  sin  2d,  p  =  sin  4:9,  p  =  sin  Qd,   .    .    . 
have  4,  8,  12,  .    .    .   loops  respectively,  each  once  drawn. 
Also  the  curve: 

p  =  sin  26  +  sin  4(9  (7) 

is  represented  by  the  heavy  curve  of  Fig.  145  as  6  varies  from  0°  to 
180°  and  by  the  dotted  curve  as  d  varies  from  180°  to  360°.     The 


/r^\rhi^ 


Fig.   146. — Curves  made  up  of  Odd  Harmonics  only,  of  Even 
Harmonics  only,  and  of  Both. 

numbered  points  1,  2,  3,  4,  .  .  .  of  Fig.  145  correspond  to  the 
similarly  numbered  points  in  Fig.  144.  The  curves  of  Fig.  144 
and  Fig.  145  correspond,  except  that  the  constant  term  was 
omitted  from  the  equation  in  constructing  Fig.  145. 

Exercises 
1.  If  /  be  the  frequency  of  the  fundamental  harmonic,  show  that: 
y  =  sin  2Tr ft  +  sin  Qirft  +  sin  lOTrft  +  •    •    • 
contains  odd  harmonics  only. 


§218]  COMPLEX  NUMBERS  365 

2.  Write  an  expression  containing  even  harmonics  only,  using  the 
frequency  /  as  in  the  last  exercise. 

3.  How  many  loops  has: 

p  =  cos  56;  p  =  sin  60; 

p  =  cos  7  ( 0  —  ^j  ;  p  =  sin  40? 

4.  In  the  diagram,  Fig.  146,  pick  out  curves  made  up  of  odd 
harmonics  only,  of  even  harmonics  only,  and  those  made  up  of  both 
odd  and  even  harmonics. 

218.*  Simple  Periodic  Variation  Represented  by  a  Complex 
Nimiber.  Fluctuating  magnitudes  exist  that  follow  the  law  of 
S.H.M,  although,  strictly  speaking,  such  magnitudes  can  be  said 
to  be  "simple  harmonic  motions"  in  only  a  figurative  sense.  For 
example  we  may  think  of  the  fluctuations  of  the  voltage  or  amper- 
age in  an  alternating  current  as  following  such  a  law.  Thus  if 
E  represent  the  electromotive  force  or  pressure  of  the  alternating 
current,  then  the  fluctuations  are  expressed  by 

E  =  Eq  sin  o)t 
or  by 

E  =  Eq  sin  2TrJt 
where  /  is  the  frequency  of  the  fluctuation.  Instead  of  S.H.M. 
such  a  variable  is  more  accurately  called  a  sinusoidal  varying 
magnitude,  although  for  brevity  we  shall  often  call  it  S.H.M. 
The  graph  in  rectangular  coordinates  of  such  a  periodic  function 
is  often  called  a  "wave,"  although  this  term  should,  in  exact 
language,  be  reserved  for  a  moving  periodic  curve,  such  as  ?/  = 
a  sin  {hx  —  kt). 
If  the  polar  representation 

p  =  a  sin  o)(t  —  ti)  (1) 

of  the  sinusoidal  varying  magnitude  be  used,  then,  as  noted  in  the 
last  section,  the  graph  of  (1)  is  a  circle  of  diameter  a  inclined  the 
angular  amount  coti  to  the  left  of  the  axis  OY,  as  is  seen  at  once 
by  calling  cot  =  6  and  o)ti  =  a  in  the  equation  of  the  circle  p  = 
a  sin  (d  —  a).  The  circle  can  be  drawn  when  the  length  and  di- 
rection of  its  diameter  are  known;  that  is,  the  circle  is  completely 
specified  when  a  and  the  direction  of  a  (told  by  a)  are  given. 
Therefore  the  simple  harmonic  motion  is  completely  symbolized 


366        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§218 

by  a  vector  OA  of  length  a  drawn  from  the  origin  in  the  direction 
given  by  the  angle  co^i;  the  direction  angle  of  the  vector  OA  is 

a  +  -  or  co^i  +  -. 

The  circle  on  the  vector  OA  is  located  or  characterized  equally 
well  if  the  rectangular  coordinates  (c,  d)  of  the  end  of  the  diameter 
of  the  circle  be  given.  But  the  complex  number  c  +  di  is  repre- 
sented by  a  vector  which  coincides  with  the  diameter  a  of  this 
circle.  Hence  we  may  represent  the  circle  by  the  complex  num- 
ber c  +  di.     Its  modulus  is  a  =    Vc^  -|-  d"^  and  its  amplitude  is 

IT  I 

0^  +  2'     Therefore  if  in  (1)  we  take  a  =    yc^  -\-  d"^,  wti  =  a  and 

the  variable  angle  o)t  =  B,  we  can  completely  determine  the  S.H.M. 
by  the  complex  number  c  +  id.  In  the  theory  of  alternating  cur- 
rents the  sinusoidal  varying  current  or  voltage  can  conveniently  be 
represented  by  a  complex  number,  and  that  method  of  repre- 
senting such  magnitudes  is  in  common  use. 

One  of  the  advantages  of  representing  S.H.M.  by  a  vector  or  by 
a  complex  number  is  the  fact  that  two  or  more  such  motions  of 
like  periods  may  then  be  compounded  by  the  law  of  addition  of 
vectors.  This  method  of  finding  the  resultant  of  two  sinusoidal 
varying  magnitudes  of  like  periods  possesses  remarkable  utility 
and  simplicity. 

To  summarize,  we  may  say: 

{a)  A  sinusoidal  varying  magnitude  is  represented  graphically 
in  polar  coordinates  by  a  vector,  which  by  its  length  denotes  the 
amplitude  and  by  its  direction  angle  with  respect  to  OY  denotes  the 
epoch  angle. 

(6)  Sinusoidal  varying  magnitudes  of  like  periods  may  be 
compounded  or  resolved  graphically  by  the  law  of  parallelogram  of 
vectors. 

If  two  sinusoidal  varying  magnitudes  of  like  periods  are  in 
quadrature  (that  is,  if  their  epoch  angles  differ  by  90°),  their  rela- 
tion, neglecting  their  epochs,  can  be  completely  expressed  by  a  sin- 
gle complex  number.     Thus  let  two  S.H.M.  in  quadrature 

j^„  =  113  sin  o}{t  -  ti)  (2) 

and 

E,  =  40  cos  o){t  -  ti)  (3) 


j218] 


COMPLEX  NUMBERS 


367 


be  represented  by  the  circles  and  by  the  vectors  marked  OEo  and 
OEc,  Fig.  147.     Call  the  resultant  of  these  Ei.     Then 

Ei  =  113  sin  0}{t  -  ti)  +  40  cos  coit  -  ti)  (4) 

=  V402  +  1132  sin  oi{t  -  ti) 
=  120  sin  o){t  -  U)  (5) 

where  oiti  is  measured  as  shown  in  Fig.  148.     Instead  of  represent- 
ing (2)  and  (3)  in  the  polar  diagram  by  OEo  and  OEc  and  their 


Fig.  147. 


-Composition  of  Two  S.H.M.  in   Quadrature  by  Law  of  Addi- 
tion of  Vectors. 


resultant  by  OEi,  we  may  represent  (2),  (3)  and  (4)  in  the  complex 
number  diagram,  Fig.  148,  by  Eo,  lE^  and  Eo  +  lEc,  respectively. 
Since  the  modulus  and  amplitude  of  Eo  +  iEc  are  -ylEo^  +  Ec^ 
and  a,  respectively,  and  since  the  epoch  angle  of  the  resultant  in 
Fig.  147  is  ct)t2  =  ooti  —  a,  we  can  state  the  resultant  as  follows: 

If  we  have  given  two  S.H.M.'s  in  quadrature  and  take  the  ampli- 
tude of  the  one  possessing  the  greater  epoch  angle  as  c  and  the 
amplitude  of  the  other  S.H.M.  as  d,  and  construct  the  complex 
number  c  -\-  di,  then  this  complex  number  c  +  di  completely 
characterizes  both  of  the  S.H.M's.  and  their  resultant.     For,  we  can 


368        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§219 

determine  the  modulus  p  and  the  amplitude  a  of  c  +  di  and  then 
if  o}ti  is  the  epoch  angle  of  the  motion  with  amplitude  c,  the  epoch 
angle  of  the  resultant  is  coti  —  a. 
If  we  consider  the  two  harmonic  motions: 
p  =  tti  sin  co(i  —  ti) 
and 

p  =  a2  sin  o){t  —  t^) 
then  if  t\  be  greater  than  U  the  first  S.H.M,  reaches  its  maximum 
value  ajter  the  second  reaches  its  maximum.    The  first  S.H.M.  is 

therefore  said  to  lag  the  amount 
(^1  —  ^2)  behind  the  second 
S.H.M.  That  is,  a  S.H.M.  rep- 
resented by  a  circle  located 
anticlockwise  from  a  second 
circle  represents  a  S.H.M.  that 
Fig.  148.-Complex  Number  /^^^  behind  the  second. 
Representation  of  the  facts  shown         ^ 

by  Polar  Diagram,  Fig.  147.  219.*  Illustration  from  Alter- 

nating Currents.  The  steady 
current  C  flowing  in  a  simple  electric  circuit  is  determined  by 
the  pressure  or  electromotive  force  E  and  the  resistance  R  ac- 
cording to  the  equation  known  as  Ohm's  law: 

^  -  R 
or, 

E  =  CR 
E  is  the  pressure  or  voltage  required  to  make  the  current  C  flow 
against  the  resistance  R.  If  the  current,  instead  of  being  steady, 
varies  or  fluctuates,  then  the  pressure  CR  required  to  make  the 
current  C  flow  over  the  true  resistance  is  called  the  ohmic  voltage 
or  ohmic  pressure.  But  a  changing  or  fluctuating  current  in  an 
inductive  circuit  sets  up  a  changing  magnetic  field  around  the  cir- 
cuit, from  which  there  results  a  counter  electromotive  force  or 
choking  effect  due  to  the  changing  of  the  current  strength.  This 
electromotive  force  is  called  the  reactive  voltage  or  reactive  pres- 
sure. The  choking  effect  that  it  has  on  the  current  is  known  as  the 
inductive  reactance.  In  case  of  a  periodically  changing  current  it 
acts  alternately  with  and  against  the  current.    Opposite  to  the 


§219]  COMPLEX  NUMBERS  369 

reactive  voltage  there  is  a  component  of  the  impressed  voltage 
that  is  consumed  by  the  reactance.     See  Fig.  149. 

The  pressure  which  is  at  every  instant  applied  to  the  circuit 
from  without  is  called  the  impressed  electromotive  force  or  vol- 
tage. Of  the  three  pressures — namely,  the  impressed  voltage,  the 
ohmic  voltage  (consumed  by  the  resistance)  and  the  reactive  vol- 
tage consumed  by  the  inductive  reactance,  any  one  may  always  be 
regarded  as  the  resultant  of  the  other  two.  Hence  if  in  a  polar 
diagram  the  pressures  be  represented  in  magnitude  and  relative 
phase  by  the  sides  of  a  parallelogram,  the  impressed  voltage  may 
be  regarded  as  the  diagonal  of  a  parallelogram  of  which  the  other 
two  pressures  are  sides.  Since,  however,  the  reactance  or  the 
counter  inductive  pressure  depends  upon  the  rate  of  change  of  the 
current,  it  lags,  in  the  case  of  a  sinusoidal  current,  90°  behind  the 
true  or  ohmic  voltage,  which  last  is  always  in  phase  with  the 
current.  The  pressure  consumed  by  the  counter  inductive  pres- 
sure therefore  leads  the  current  by  90°.  Thus,  in  the  language  of 
complex  numbers 

Ei  =  Eo-\-  lEo  (1) 

in  which 

Ei  =  impressed  pressure 

Eo  =  ohmic    pressure,    or    pressure    consumed   by    the 
resistance 

Ec  =  counter  inductive  pressure,  or  the  pressure  con- 
sumed by  reactance 

It  is  found  that  the  counter  inductive  pressure  depends  upon  a  con- 
stant of  the  circuit  L  called  the  inductance  and  upon  the  angular 
velocity  or  frequency  of  the  alternating  impressed  pressure,  so  that: 

Ec  =  2TrfLC  =  coLC 

Hence  (1)  may  be  written: 

Ei  =  RC  +  i2TfLC  (2) 

=  RC  +  loiLC  (3) 

The  modulus  of  the  complex  number  on  the  right  of  this  equation  is 


C  yjR2  +  w^L' 

24 


370        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§219 


Considering,  then,   merely  the  absolute 
pressure  and  current,  we  may  write: 


\C\  = 


V/22   4_   ^22,2 


value  \Eo\   and  \C\  of 


(4) 


From  the  analogy  of  this  to  Ohm's  law: 

E 
R 


C  = 


the  denominator  Vi?^  +  co^L^  is  thought  of  as  limiting  or  restrict- 
ing the  current  and  is  called  the  impedance  of  the  circuit. 

Let  there  be  a  condenser  in  the  circuit  of  an  alternator,  but  let 
the  circuit  be  free  from  inductance.  Then  besides  the  pressure 
consumed  by  the  resistance,  an  additional  pressure  is  required  at 
any  instant  to  hold  the  charge  on  the  condenser.  If  K  be  the  ca- 
pacity of  the  condenser,  it  is  found  that  that  part  of  the  pressure 

C  C 

consumed  in  holding  the  charge  on  the  condenser  is  ft— ^7  or  ~r> 

ZtjK         a)K 

and  is  in  phase  position  90°  be- 
hind the  current  C.  The  chok- 
ing effect  of  this  on  the  current 
may  be  called  the  condensive 
reactance.  When  a  condenser 
is  in  the  circuit  in  addition  to  in- 
ductance, the  total  pressure  con- 
sumed by  the  reactance  has  the 
form: 

and  the  complex  number  that  symbolizes  the  vector  is 

Ei  =^RC  +  i2TfCL  -  2^jr^  (5) 

(see  Fig.  149). 

Further  illustrations  of  the  applications  of  complex  numbers  to 
alternating  currents  is  out  of  place  in  this  book.  The  illustrations 
are  merely  for  the  purpose  of  emphasizing  the  usefulness  of  these 
numbers  in  applied  science.  An  interesting  application  of  the  use 
of  complex  numbers  to  the  problem  of  the  steam  turbine  will  be 
found  in  Steinmetz's  "Engineering  Mathematics,"  page  33. 


Fig.  149. — Complex  Number  Dia- 
gram'of  Equation  5,   §219 


§220] 


COMPLEX  NUMBERS 


371 


(2  A-  2t)  ( VT+  i) 


Exercises 

1.  Draw  the  polar  diagram  and  complex  number  representation 
of  Ei  if  i?  =  5,  C  =21,  /  =  60,  L  =  0.009,  K  =  0.005. 

2.  Draw  a  similar  diagram   if  Ei   =  100,  Eo  =  90,  /  =  40,  L  = 
0.008,  K  =  0.003. 

220.  Product  of  Complex  Numbers.     The  product  of  two  or 
more  complex  numbers  is  a  complex  number  whose  modulus  is  the 
prodwct  of  the  moduli  and  whose 
amplitude  is  the  sum  of  the  am- 
plitudes of  the  complex  numbers. 
Let  the  complex  numbers  be: 
z\  =  xi  +  yii 

=  ri  (cos  ^1  +  i  sin  di) 
Z2  =  Xi  -\-  yii 

=  rz  (cos  02  +  i  sin  ^2),  etc. 
By  actual   multiplication,  we 
get: 

21^2  =  riri  [(cos  ^1  cos  62 

—  sin  ^1  sin  ^2) 

+  (sin  61  cos  02  +  cos  01  sin^z)^] 

=  riri  [cos  (01  +  02) 

+  i  sin  (01  +  02)] 

Whence  it  is   seen  that  rir2  is 

the  modulus  of  the  product  and 

(01  +  02)  is  the  amplitude. 

The  above  theorem  is  illustrated 
by  Fig.  150.  If  the  two  given  complex  numbers  be  represented 
by  their  vectors  OPi  and  OP2,  their  product  will  be  represented 
by  the  vector  OP3  whose  direction  angle  is  the  sum  of  the  ampli- 
tudes of  the  two  given  factors,  and  whose  length  OP 3  is  the  product 
of  the  lengths  OPi  and  OP 2. 

The  figure  represents  the  product  (2  +  2i)(V3  +  i).    Expressed 
in  terms  of  modulus  and  amplitude  these  may  be  written: 
V3  +    i  =  2  (cos  30°  +  i  sin  30°) 
2  4-  2i  =  2V2(cos  45"  +  i  sin  45°) 
Hence  ri  =2,      rz  =  2V2,       di  =  30°,       62  =  45° 
Therefore:        (2  +  2i)(V3  +  i)  =  4\/2  (cos  75°  +  i  sin  75°) 


Fig. 


150.— Product  of  Two  Com- 
plex Numbers. 


372        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§221 

Exercises 

Find  the  moduli  and  amplitudes  of  the  following  products,  and 
construct  the  factors  and  products  graphically.  Take  a  positive  angle 
for  the  amplitude  in  every  case. 

1.  (1  +V3t)(2V3  +2^). 

2.  (2  +  fV3i)(2  +2i). 

3.  (V3+3*)(2-20. 

4.  (l+t)2._ 

5.  (2  -  2V30 W3  +  3i). 

6.  (1  -  i)\ 

7.  (1  +iy{l  -i)\ 

8.  2(cos  15°  +  i  sin  15°)  X  3(cos  25°  +  i  sin  25°). 

Find  numerical  result  by  use  of  slide  rule  or  trigonometric  tables. 

9.  2(cos  10°  +  i  sin  10°)  X  (1/3) (cos  12°  +  i  sin  12°)  X  6(cos  8° 
+  i  sin  8°). 

10.  Find    the    value    of    4V2(cos  75°  +  i  sin    75°)  -^  (V3  +  t). 

221.  Quotient  of  Two  Complex  Numbers.  The  quotient  of 
two  complex  numbers  is  a  complex  number  whose  modulus  is  the 
quotient  of  the  moduli  and  whose  amplitude  is  the  difference  of  the 
amplitudes  of  the  two  complex  numbers.  Let  the  complex  numbers 
be: 

Zi  =  xi  +  yii  =  ri(cos  ^i  +  i  sin  ^i) 

Z2  =  X2  -^  Vii  =  ^2  (cos  62  +  i  sin  ^2) 
We  have: 

2i  _  ri(cos  ^1  +  ^'sin  gi)(cos  62  —  ism  62) 
22  •"  ?'2(cos  d2-\-i  sin  g  2)  (cos  62  —  i  sin  62) 

^  ri[cos  (^1  -  ^2)  +  i  sin  (^1  -  ^2)] 

~  r2  (cos2  02  +  sin^^a) 

=  -[cos  (^1  -  62)  +  i  sin  (^1  -  ^2)] 
r2 

Whence  it  is  seen  that  —  is  the  modulus  of  the  quotient  and 
r2 

(di  —  6 2)  is  the  amplitude. 

In  Fig.  151,  the  complex  number  represented  by  the  vector  OPi 

when  divided  by  the  complex  number  represented  by  OP2  yields 

the  result  represented  by  OP3,  whose  length  ri/r2  is  found  by  dividing 

the  length  of  OPi  by  the  length  of  OP2,  and  whose  direction  angle 


§222] 


COMPLEX  NUMBERS 


373 


is  the  difference  {di  —  62)  of  the  amplitudes  of  OPi  and  OP2.     The 
figure  is  drawn  to  scale  for  the  case: 

5  (cos  60°+  i  sin  60°) 
2(cos20°+  i  sin  20°) 


(2.5)  (cos  40°  +  i  sin  40°) 


Exercises 

Find  the  quotient  and  graph  the  result  in  each  of  the  following 
exercises.  Always  take  amplitudes  as  positive  angles  and  if 
02  <  di,  take  di  +  360°  instead  of  di. 


Fig.  151. — Quotient  of  Two  Complex  Numbers. 

1.  (1  +  Vsi)  ^  (2  +  V2i). 

2.  (I  +  ^  Vsi)  -^  (  V2  -  V2i). 

3.  (3  Vs  -  3i)  -^  (  -  1  +  ^lsi). 

4.  (1  -  slsi)  ^  i. 

6.     2  (cos  36°  +  i  sin  36°)  ^  5  (cos  4°  +  i  sin  4°). 
6.  12(cos  48°  +  i  sin  48°)  ^  [2(cos  15°  +  i  sin  15°) 

3  (cos  9°  +  i  sin  9°)]. 


7. 


[4+  (4/3)V3z](2  +  2V3i) 


8  +  8i 
8.  Express  in  terms  of  a,  h,  c,  d  the  amplitude  of  (a  +  hi)  -^  (c  +  di) . 

222.  De  Moivre*s  Theorem.     As  a  special  case  of  §220  consider 
the  expression: 

(cos  e  +  i  sin  B)"" 


374        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§222 

This  being  the  product  of  n  factors  like  (cos  B  -\-  i  sin  6),  we  write, 

by  means  of  §220: 

(cos  ^ -h  i  sin0)(cos  ^  4- z  sin^)   .    .•  . 

=  [cos  ((9  +  (9  +  .    .    .   )  +  i  sin(0  +  ^  +  •    •    .   )t 
or: 

(cos  d  +  i  sin  BY  =  (cos  nB  -{-  i  sin  nB)  (1) 

which  relation  is  known  as  De  Moivre's  theorem. 

De  Moivre's  theorem  holds  for  fractional  values  of  n.     For,  first 
consider  the  expression: 

(cos^  +  isin^)'/' 
where  the  power  1/t  of  cos  B  -\-  i  sin  B  is,  by  definition,  an  oper- 
ator such  that  its  tih  power  equals  cos  B  -\-i  sin  B. 

a 

Put    B  =  t(f),  so  that  0  =  7 
Then:  (cos  B  +  ^  sin  B)'^'  =  (cos  tcf)  +  i  sin  «(/>)'/' 

=  [(cos  0  +  1^  sin  0)']'/'  by  (1) 
=  cos  0  +  *  sin  0 

=  COS  7  -\-  i  sin  -  (2) 

t  I 

o 

Next  consider  the  case  in  which  n  =  . .     We  know: 
(cos  B  +  1  sin  B)'^'  =  [(cos  ^  +  ^  sin  B)")]'^' 

=  (cossB +  i  sin  sB)'^' by  (1) 

=  cos  y  +  i  sin  y  by  (2)  {3) 

Likewise  the  theorem  may  be  proved  for  negative  values  of  n. 
The  following  examples  illustrate  the  application  of  De  Moivre's 
theorem. 

(1)  Find  (3  +  i  ^lsy. 

write:  3  +  W3  =  2  V3(cos  30°  +  i  sin  30°) 

Then,  by  De  Moivre's  theorem: 

(3  +  W3)4  =  144 (cos  120°  +  i  sin  120°) 
=  144  (  -  1/2  +  Wsi) 

=  -72  +  72  yisi 

(2)  Find  (2  +  2i)ii. 


§223]  COMPLEX  NUMBERS  375 

Write:     2  +  2i  in  the  form: 

2  +2z  =  2  V2(W2  +  W2^) 
(2  +  2^)"  =  (2  V2)ii(cos  45°  +  i  sin  45°) ii 
=  (2  V2)  11(008  495°  +  i  sin  495°) 
=^  (2  V2)"(cos  135°  +  i  sin  135°) 

=  (2  V2)iH  -  W2  +  ^  V2i) 
=  2i6(-l+i) 

Exercises 

Evaluate  the  following  by  De  Moivre's  theorem,  using  trigonometric 
table  or  slide  rule  when  necessary. 

1.  (8  +  8  ^J3^y\ 

2.  V27(cos75°  -i  sin  75°). 

3.  ^125  i. 

4.  [cos  9°  +  i  sin  9°]". 
6.  (3  +  ^lsi)'. 

'       6.  [1/2  +  (1/2)  yl3i]K 

7.  (1  +  iy. 

8.  (-2  +2i)H. 

9.  1(1/2)  VS  -  (l/2)i]5.  

10.  Find  value   of  (  -  1  +   V  -  3)^  +  (  -  1  -  V  -  3)^   by    De 
Moivre's  theorem. 

11.  Find  the  value  of  x^  -  2x  +  2  for  a;  =  1  +  i. 

12.  Ifii  =  -  1/2  +  (1/2)   V  -  3  and  J2  =  -1/2  -  (1/2)  V  -  3, 
show    that   j  i'  =  1,     J2^  =  1,     ii^  =  , 

223.  The  Roots  of  Unity.     Unity  may  be  written: 
1  =  cos  0  +  i  sin  0 
1  =  cos  2'K  -\- 1  sin  27r 
1  =  cos  47r  +  i  sin  47r 
1  =  cos  Gtt  +  i  sin  Gtt 


and  so  on.     By  De  Moivre's  theorem  the  cube  root  of  any  of  these 


376        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§223 

is  taken  by  dividing  the  amplitudes  by  3.     Therefore,  from  the 
above  expressions  in  turn  there  results: 

'^I  =  cos  0  +  i  sin  0  =  1 

^I  =  cos  (27r/3)  +  i  sin  (27r/3)  =  cos  120°  +  i  sin  120° 

=  -1/2 +  1(1/2)  V  3 
Vl  =  cos  (47r/3)  +  I  sin  (47r/3)  =  cos  240°  +  i  sin  240° 

=  -1/2  -  iXl/2)V3 
Vi  =  cos  Gtt  /3  +  i  sin  67r  /3  =  same  as  first,  etc. 


Fig.   152.— The  Cube  Roots  of  Unity. 


Therefore  there  are  three  cube  roots  of  unity.  Since  these  are  the 
roots  of  the  equation  x^  —  1  =  0,  they  might  have  been  found  by 
factoring,  thus: 

x3  -  1  =  (x  -  l)(a;2  +  x+  1) 

=  (a;-l)(a;+l/2  +  i^/3^)(a:  +  l/2-iv/3^) 
The  three  roots  of  unity  divide  the  angular  space  about  the  point 
0  into  three  equal  angles,  as  shown  in  Fig.  152.  In  the  same 
way,  it  can  be  shown  that  there  are  four  fourth  roots,  five  fifth 
roots,  etc.,  of  unity  and  that  the  vectors  representing  them  have 
modulus  1  and  amplitudes  that  divide  equally  the  space 
about  0. 


§223]  COMPLEX  NUMBERS  377 

To  find  all  of  the  roots  of  any  complex  number,  proceed  as  in  the 
following  illustrative  examples. 

(1)  Find  VV3  +  Si. 
Write  Vs  +  3i  in  the  form: 

V3  +  3i  =  2V3(cos  60°  +  i  sin  60°) 
Hence,  by  De  Moivre's  theorem: 

(V3  +  31)'"^  =  \/l2  (cos  30°  +  i  sin  30°) 
=  \/l2  [(1/2)  V3  +  (l/2)^] 
=  (1/2)V^108  +  (l/2)\/l2  i 
A  second  root  can  be  found  by  writing : 

V3  +  3^  =  2V3[cos  (60°  +  360°)  +  i  sin  (60°  +  360°)] 
since  adding  a  multiple  of  360°  to  the  amplitude  does  not  change  the 
value  of  the  sine  and  cosine.     In   applying  De   Moivre's  theorem 
there  results: 

(Vs  +  3i)'/^  =  \/l2  (cos  210°  +  i  sin  210°) 
=  a/i2[  -  (1/2)V3  -  (l/2)i] 

(2)  Find  the  cube  root  of  —  V2  +  V2i. 
We  write: 

-  V2  +  W2  =  2 (cos  135°  +  i  cos  135°) 

=  2[cos  (135°^+'w360°)>  ^  cos  (135°  +  n360°)] 

in  which  n  is  any  integer.     Hence: 

(-  V2  +  ^  V2')^/^  =  '^2[cos  (45°  +  nl20°)  +  i  sin  (45°  +  n  120°)]^ 
=  i^2(cos  45°  +  i  sin  45°)     forn  =  0 
=  'i^2(cos  165°  +  i  sin  165°)  forn  =  1 
=  V2(cos  285°  +  i  sin  285°)  f or  n  =  2 

These  are  the  three  cube  roots  of  the  given  complex  number.     For 

n  =  3  the  first  root  is  obtained  a  second  time. 

Exercises 

Find  all  the  indicated  roots  of  the  following: 

1.  (8  +  sylsi)]^. 

2.  V27(cos  75°  -  i  sin  75°). 

3.  '^lL25i. 


378        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§224 

4.  (  -  2  +  2i)y\ 
6.  (2  +  2i)^^. 

6.  32  ^1 

7.  V^sK 

8.  Find  to  four  places  one  of  the  imaginary  7th  roots  of   +  1. 
Note:     Cos  51°  25.7'  +  i  sin  51°  25.7'  =  0.6235  +  0.7818  i. 

224.  Inverse  Functions.  The  exponent  ( —  1)  attached  to  a  sym- 
bol of  operation  signifies  the  "undoing"  of  the  operation  denoted 
by  the  symbol  of  operation.  The  number  of  different  operations 
in  mathematics  is  an  even  number;  that  is,  for  every  operation  we 
define,  we  may,  and  usually  do,  define  the  operation  that  ''un- 
does" the  given  operation.  Thus  if  we  define  addition,  we  at  once 
follow  it  by  defining  the  undoing  of  addition,  or  subtraction;  if 
we  define  multiplication,  we  follow  it  with  the  concept  of 
the  undoing  of  multiplication,  or  division;  if  we  define  in- 
volution, or  the  raising  to  powers,  we  also  define  the  undoing  of  this 
operation,  namely  evolution,  or  the  extraction  of  roots.  The 
second  of  each  of  these  pairs  of  operations  is  called  the  inverse 
of  the  first  operation,  and  vice  versa. 

The  exponent  ( —  1)  attached  to  any  symbol  of  operation  is  defined 
to  mean  the  inverse  of  the  operation  called  for  by  the  symbol  to 
which  it  is  attached.  Thus  2~^  is  not  a  doubter;  the  operation 
called  for  is  the  "undoing''  of  doubling,  or  halving.  The  symbol 
log-i  X,  read  the  ''anti-logarithm  of  x,"  calls  for  the  number  of 
which  X  is  the  logarithm.  Thus,  if  the  base  be  10,  log-^  2  =  100, 
log-i  3  =  1000,  log-i  0.3010  =  2,  log-i  1  =  10,  log-i  0=1,  etc. 
Since  "log"^"  is  the  symbol  of  undoing  the  operation  indicated 
by  "log,"  the  double  symbol  (log~i  log)  must  leave  the  oper- 
and unchanged.  The  operator  that  leaves  an  operand  unchanged 
is  unity.  Hence  a  double  symbol  like  (log-^  log)  can  always  be 
replaced  by  1;  thus  log-^  log  467  =  467;    also  log  log-^  467  =  467. 

Likewise  3-1-3-1  =  1;  (V3)-i-(V3)-l  =  1,  etc. 

An  important  use  of  the  pre3ent  notation  is  in  the  symbols  sin-^  x, 
cos-^  X,  tan  -^  x,  etc.,  used  in  §70.  These  are  read  "anti-sine  of 
X,"  *'anti-cosine  of  a,"  etc.,  or  "the  angle  whose  sine  is  x,"  ''the 
angle  whose  cosine  is  x,"  etc.     Thus  sin~^  (1  /2)  =  30°,    tan-^ 


§224]  COMPLEX  NUMBERS  379 

=  45°,  cos-^  0  =  TT  /2,  etc.  Note  that  log-^  x  must  be  care- 
fully distinguished  from  (log  .t)~^,  which  means  1/log  x;  simi- 
larly, sin~^  X  must  be  distinguished  from  (sin  x)~^.  A  notation 
like  log  x-^  is  ambiguous,  and  should  never  be  used. 

If  we  write  r  =  cos  0,  y  =  log  x,y  =  tan  x,  the  same  func- 
tional relations  may  be  expressed  in  the  inverse  notation  by 
6  =  cos-i  r,  X  =  log-i  y,  X  =  tan-i  y.  Thus  y  =  a'',  x  =  logoi/, 
y  =  logo~^  X,  and  y  =  expoX  are  four  ways  of  expressing  the 
same  relation  between  x  and  y. 

Any  relation  expressed  by  means  of  the  direct  functions  may  also 
be  expressed  in  terms  of  the  inverse  functions.     Thus  we  know: 

log  (xy)  =  log  X  +  log  y  (I) 

Let  log  X  =  a,  log  y  =  h,  then  it  follows  that: 

x  =  log~i  a,  y  =  log"^  b 

Hence  (1)  becomes: 

log  (log"^  a  log"i  b)  =  a  -\-  b 
or: 

log-i  a  log-i  b  =  log-i  (a  +  b)  (2) 

Likewise  consider: 

sin  (a  +  /3)  =  sin  a  cos  /3  +  cos  a  sin  /3  (3) 

Let  sin  a  =  a  and  sin  /3  =  6 

then:  a  =  sin~^  a,  0  =  sin'^b 

Also  since  sin  a  =  a 


cos  a  =    Vl  —  o^ 
Likewise: 

cos  /3  =    Vl  -  62 

Hence  (3)  may  be  written:  

sin  (sin-la  +  sin"'  b)  =  aVl  -  b^  +  bVl  -  a^ 

or:  .     

sin-i  a  +  sin-i  b  =  sin"!  (aVl  -  6^  +  b  Vl  -  a«) 

Since  there  are  many  angles  whose  sine  is  equal  to  a  given 
number  x,  it  is  desirable  to  specify  by  definition  which  angle 
is  meant.     The  following  conventions  are  therefore  useful: 

sin-i  X  means  the  angle  between  —  90°  and  +  90°  whose 
sine  is  x. 


380        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§224 

cos~^  X     means    the    angle    between  0°    and    180°     whose 
cosine  is  x. 

tan~^  X  means  the  angle  between  —  90°  and  +  90°  whose 
tangent  is  x. 

Exercises 

1.  Show  that  sin-i  (1/2)  +  sin-i  ■\ll/2  =  57r/12. 

2.  Show  that  sin~^  x  +  cos"^  x  +  cos~^  x  =  ■3r/2. 

3.  Is  there  any  difference  between  the  graph  oi  y  =  f{x)  and  the 
graph  of  X  =  f~^(y)? 

4.  Prove  that  tan-^a;  +  tan"!  (1/rc)  =  7r/2.i 

6.  Find  the  value  of  x  in  the  equation  sin"^  x  +  sin~^  2x  =  tt/S. 

6.  If /(re)  =x',  find/-Hx). 

Let  y  =  f(x)  =  x^.     Then  x  =  f~'^{y)  =  ^/y.     Hence  if  /"Hz/)  = 
y/y,  then  /~K^)  =  \/^- 

7.  If/(x)  =e%find/-i(a;). 

8.  What  is  the  inverse  of  f{e)  =1-0? 
Let  y  =  f{d),  so  that  0  =  j-'^{y),  etc. 

9.  Show  that  the  function 

X  +  1 

y  = 7 

X  —  1 

is  its  own  inverse. 

iThe  symbol  (  =  )  may  here  be  interpreted  as  meaning  "congruent  to." 


CHAPTER  XII 
LOCI 

225.  Parametric  Equations.  The  equation  of  a  plane  curve  is 
ordinarily  given  by  an  equation  in  two  variables,  as  has  been  amply 
illustrated  by  numerous  examples  in  the  preceding  chapters.  It 
is  obvious  that  a  curve  might  also  be  given  by  two  equations  con- 
taining three  variables,  for  if  the  third  variable  be  eliminated  from 
the  two  equations,  a  single  equation  in  two  variables  results. 
When  it  is  desirable  to  describe  a  locus  by  means  of  two  equations 
in  three  variables  the  equations  are  known  as  parametric  equations, 
as  has  already  been  explained  in  §74.  Two  of  the  variables  usu- 
ally belong  to  one  of  the  common  coordinate  systems  and  the  third 
is  an  extra  variable  called  the  parameter.  In  applied  science  the 
variable  time  frequently  occurs  as  a  parameter. 

The  parametric  equations  of  the  circle  have  already  been  written. 
They  are: 

a;  =  a  cos  ^  (1) 

y  =  asind 
where  the  parameter  6  is  the  direction  angle  of  the  radius  vector 
to  the  point  {x,  y).    Likewise  the  parametric  equations  of  the 
ellipse  have  been  written: 

X  =  a  cos  B  (2) 

2/  =  6  sin  0 
and  those  of  the  hyperbola  have  been  written: 

a;  =  a  sec  ^  (3) 

2/  =  6  tan  d 
In  harmonic  motion,  the  ellipse  was  seen  to  be  the  resultant  of 
the  two  S.H.M.  in  quadrature: 

X  =  a  cos  oit  (4) 

2/  =  6  sin  (i3t 
Here  the  parameter  t  is  time. 

226.  Problems  in  Loci.  It  is  frequently  required  to  find  the 
equation  of  a  locus  when  a  description  of  the  process  of  its  genera- 
tion is  given  in  words,  or  when  a  mechanism  by  means  of  which  the 

381 


382        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§226 


curve  is  generated  is  fully  described.  There  is  only  one  way  to 
gain  facility  in  obtaining  the  equations  of  curves  thus  described, 
and  that  is  by  the  solution  of  numerous  problems.  Sometimes 
it  is  best  to  seek  the  parametric  equations  of  the  curve,  but 
sometimes  the  ordinary  polar  or  Cartesian  equation  can  be  ob- 
tained directly.     The  following  problems  are  illustrative: 

(1)  A  straight  line  of  constant  length  a  -\-  b  moves  with  its  ends 
always  sliding  on  two  fixed  lines  at  right  angles  to  each  other.  Find 
the  equation  of  the  curve  described  by  any  point  of  the  moving 
line.     (See  §75.) 


Fig.   153. — Generation  of  So-called  "Elliptic  Motion." 

In  Fig.  153,  let  AB  be  the  line  of  fixed  length,  and  let  it  so  move 
that  A  remains  on  the  x-axis  and  B  remains  on  the  ^/-axis.  Let  any 
point  of  this  line  be  P  whose  distance  from  A  is  6  and  whose  distance 
from  B  is  a.  If  the  angle  X'AB  be  called  B,  thenPD,  the  ordinate  of 
P,  is: 

y  =  6  sin  0 
and  OD,  the  abscissa  of  P,  is: 

X  =  a  cos  d 
Therefore  P  describes  an  ellipse  of  semi-axes  a  and  h. 

(2)  A  circle  rolls  without  slipping  within  a  circle  of  twice  the 
diameter.  Show  that  any  point  attached  to  the  moving  circle^ 
describes  an  ellipse. 


1  "Circle"  is  here  used  in  the  sense  of  a   "disc' 
the  sense  of  a  "circumference." 


or  circular  area  and  not   in 


§226]  LOCI  383 

Draw  the  smaller  rolling  circle  in  any  position  within  the  larger 
circle,  and  call  the  point  of  tangency  T,  as  in  Fig.  153.  Since 
the  smaller  circle  is  half  the  size  of  the  larger  circle,  the  smaller 
circle  always  passes  through  0,  and  the  line  joining  the  points  of 
intersection  of  the  small  circle  with  the  coordinate  axes  is,  for  all 
positions,  a  diameter,  since  the  angle  AOB  is  a  right  angle. 

If  we  can  prove  that  the  arc  AT  =  the  arc  HT  for  all  positions 
of  T,  then  we  shall  have  shown  that  as  the  small  circle  rolls  from  an 
initial  position  with  point  of  contact  at  H,  the  end  A  of  the  diameter 
AB  slides  on  the  line  OX.  Since  B  lies  on  OF  and  since  AB  is  of 
fixed  length,  this  proves  by  problem  (1)  that  any  point  of  the  small 
circle  lying  on  the  particular  diameter  AB  describes  an  ellipse. 

To  prove  that  arc  AT  =  arc  HT,  we  have  that  the  angle  HOT 
arc  HT 
is  measured  in  radians  by  — 7)ir~      The  angle  AO'T  is  measured  in 

nrp   A  T 

radians  by      q>j[  '     Since  Z  AO'T  =  2 /.HOT,  we  have 

arc  AT  _     arc  HT 
O'A    ~  ^      OH 
But,  OH  =  20' A.     Hence  ssg  AT  =  arc  HT. 

We  can  now  prove  that  any  other  point  of  the  rolling  circle  de- 
scribes an  ellipse.  Let  any  other  point  be  Pi.  Through  Pi  draw 
the  diameter  JO'K.  The  above  reasoning  applies  directly,  replacing 
A  by  J  and  H  by  N. 

It  is  easy  to  see  that  all  points  equidistant  from  the  center  such 
as  the  points  P,  Pi,  of  the  small  circle,  describe  ellipses  of  the  same 
semi-axes  a  and  h,  but  with  their  major  axes  variously  inclined  to 
OH. 

(3)  Determine  the  curve  given  by  the  parametric  equations: 

X  =  a  cos  2o}t  (1) 

y  =  a  sin  cat  (2) 

To  eliminate  t,  the  first  equation  may  be  written 

X  =  ail  -2  sin2  ut)  (3) 

y 

From  the  second  equation,  sin  cot  =  —     Substituting  for  sin  (at  in 
(3), 

or, 

a         a^ 


384        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§226 

This  curve  is  the  parabola  y"^  =  mx,  the  special  location  of  which  the 
student  should  describe. 

(4)  Construct  a  graph  such  that  the  increase  in  y  varies  directly 
as  X. 

If  y  varied  directly  as  x,  then  y  would  equal  kx,  where  k  is  any 
constant.  In  the  given  problem  the  increase  in  y  (and  not  y  itself) 
must  vary  in  this  manner.  Let  the  initial  value  of  y  be  represented 
by  ?/o.  Then  the  gain  or  increase  of  y  is  represented  hy  y  —  yo. 
Hence,  by  the  problem: 

y  —  yo  =  kx  (1) 

Since  yo  is  a  constant,  (1)  is  the  equation  of  the  straight  line  of  slope 
k  and  intercept  on  the  y-axis  =  yo,  which  ordinarily  would  be  written 
in  the  form : 

y  =  kx  +  yo 

(5)  Express  the  diagonal  of  a  cube  as  a  function  of  its  edge,  and 
graph  the  function. 

If  the  edge  of  the  cube  be  x,  its  diagonal  is  Vx^  +  x'-{-  x^ or  xVs. 
If  the  diagonal  be  represented  by  y,  we  have  y  =  yJSx,  which  is  a 
straight  line. 

(6)  A  rectangle  whose  length  is  twice  its  breadth  is  to  be  in- 
scribed in  a  circle  of  radius  a.  Express  the  area  of  this  rectangle 
in  terms  of  the  radius  of  the  circle. 

Let  the  rectangle  be  drawn  in  a  circle  whose  equation  is  x^  -\-  y^  =  a^. 
At  a  corner  of  the  rectangle  we  have  x  =  2y.  The  area  A  of  the 
rectangle  is  4:xy,  or  since  x  =  2y,  is  ^y^.  From  the  equation  of  the 
circle  we  obtain  42/^  +  2/^  =  a^  or  y"^  =  a'^/b.     Hence: 

A  =  (8/5)a2 

If  A  and  a  be  graphed  as  Cartesian  variables,  the  graph  is  a  parabola. 

(7)  A  rectangle  is  inscribed  in  a  circle.  Express  its  area  as  a 
function  of  a  half  of  one  side. 

Here,  as  above: 

A  =  4iXy  =  4xVa2  —  x^ 
The  student  should  graph  this  curve,  for  which  purpose  a  may  be 
put  equal  to  unity.  First  draw  the  semicircle  y  =  y  a^  —  x^. 
For  a;  =  1/5  take  one-fifth  of  the  ordinate  of  this  semicircle.  For 
x  =  2/5  take  two-fifths  of  the  ordinate  of  the  semicircle,  and  so  on. 
The  curve  through  these  points  is  y  =  x^Ja^  —  x^,  from  which 
y  =  4a;Va2  —  x^  can  be  had  by  proper  change  in  the  vertical  unit  of 
measure. 


§226]  LOCI  385 

Exercises 

1.  In  polar  coordinates,  draw  the  curves : 

r  =  2  cos  0  r  =  2  cos  ^  +  1 

r  =  2  cos  a  -  1  r  =  2  cos  0  +  3. 

2.  A  curve  (polar  coordinates)  passes  through  the  point  (1,  1). 
(This  means  the  point  whose  coordinates  are  one  centimeter,  and  one 
radian.)  Starting  at  this  point,  a  point  moves  so  that  the  radius 
vector  of  the  point  is  always  equal  to  the  vectorial  angle.  Sketch 
the  curve.     Write  the  polar  equation  of  the  curve. 

3.  A  point  moves  so  that  one  of  its  polar  coordinates,  the  radius 
vector,  varies  directly  as  the  other  polar  coordinate,  the  vectorial 
angle.  Write  the  polar  equation  of  such  a  curve.  Does  the  curve 
go  through  the  point  (1,  1)? 

4.  A  polar  curve  is  generated  by  a  point  which  starts  at  the  point 
(1,  2)  and  moves  so  that  the  increase  in  the  radius  vector  always 
equals  the  increase  in  the  vectorial  angle.  Write  the  equation 
of  the  curve. 

5.  A  polar  curve  is  generated  by  a  point  which  starts  at  the  point 
(1, 2)  and  moves  so  that  the  increase  in  the  radius  vector  varies  directly 
as  the  increase  in  the  vectorial  angle.     Write  the  equation  of  the  curve. 

6.  A  ball  is  thrown  from  a  tower  with  a  horizontal  velocity  of  10 
feet  per  second.  It  falls  at  the  same  time  through  a  variable  distance 
given  by  s  =  16.1^2,  where  t  is  the  elapsed  time  in  seconds  and  s  is 
in  feet.     Find  the  equation  of  the  curve  traced  by  the  ball. 

7.  The  point  P  divides  the  line  AB,  of  fixed  length,  externally  in 
the  ratio  a  :h,  that  is,  so  that  PA/PB  =  a/b.  If  the  line  AB  move 
with  its  end  points  always  remaining  on  two  fixed  lines  OX  and  0  Y 
at  right  angles  to  each  other,  then  P  describes  an  ellipse  of  semi-axes 
a  and  h. 

8.  If  in  the  last  problem  the  lines  OX  and  OY  are  not  at  right 
angles  to  each  other,  the  point  P  still  describes  an  ellipse. 

9.  A  point  moves  so  as  to  keep  the  ratio  of  its  distances  from 
two  fixed  lines  AC  and  BD  constant.  Prove  that  the  locus  consists 
of  four  straight  lines. 

10.  A  sinusoidal  wave  of  amplitude  6  cm.  has  a  node  at  +  5  cm. 
and  an  adjacent  crest  at    +  8  cm.     Write  the  equation  of  the  curve. 

11.  The  velocity  of  a  simple  wave  is  10  meters  per  second.  The 
period  is  two  seconds.     Find  the  wave  length  and  the  frequency. 

12.  A  polar  curve  passes  through  the  point  (1,  1)  and  the  radius 
vector  varies  inversely  as  the  vectorial  angle.  Plot  the  curve  and 
write  its  equation.    Consider  especially  the  points  where  the  vectorial 

25 


386     .  ELEMENTARY  MATHEMATICAL  ANALYSIS      [§227 

angle  becomes  infinite  and  where  it  is  zero.     Sketch  the  same  func- 
tion in  rectangular  coordinates. 

13.  Rectangles  are  inscribed  in  a  circle  of  radius  r.  Express  by- 
means  of  an  equation  and  plot :  (a)  the  area,  and  (6)  the  perimeter 
of  the  rectangles  as  a  function  of  the  breadth. 

14.  Right  triangles  are  constructed  on  a  line  of  given  length  h 
as  hypotenuse.  Express  and  plot:  (a)  the  area,  and  (6)  the  per- 
imeter as  a  function  of  the  length  of  one  leg. 

16.  A  conical  tent  is  to  be  constructed  of  given  volume,  V.  Express 
and  graph  the  amount  of  canvas  required  as  a  function  of  the  radius 
of  the  base. 

16.  A  closed  cylindrical  tin  can  is  to  be  constructed  of  given  volume, 
V.  Plot  the  amount  of  tin  required  as  a  function  of  the  radius  of  the 
can. 

17.  A  rectangular  water-tank  lined  with  lead  is  to  be  constructed 
to  hold  108  cubic  feet.  It  has  a  square  base  and  open  top.  Plot 
the  amount  of  lead  required  as  a  function  of  the  side  of  the  base. 

18.  An  open  cylindrical  water-tank  is  to  be  made  of  given  volume, 
V.  The  cost  of  the  sides  per  square  foot  is  two-thirds  the  cost  of 
the  bottom  per  square  foot.  Plot  the  cost  as  a  function  of  the 
diameter. 

19.  An  open  box  is  to  be  made  from  a  sheet  of  pasteboard  12  inches 
square,  by  cutting  equal  squares  from  the  four  corners  and  bending 
up  the  sides.  Plot  the  volume  as  a  function  of  the  side  of  one  of  the 
squares  cut  out. 

20.  The  illumination  of  a  plane  surface  by  a  luminous  point  varies 
directly  as  the  cosine  of  the  angle  of  incidence,  and  inversely  as  the 
square  of  the  perpend'cular  distance  from  the  surface.  Plot  the 
illumination  of  a  point  on  the  floor  10  feet  from  the  wall,  as  a  func- 
tion of  the  height  of  a  gas  burner  on  the  wall. 

21.  Using  the  vertical  distances  between  corresponding  points  on 
the  curves  y  =  sm  t  and  y  =  —  sin  ^  as  ordinates  and  the  vertical 
distances  between  corresponding  points  of  y  =  2t  and  y  —  t^  as 
abscissas,  find  the  equation  of  the  resulting  curve. 

227.  Loci  Defined  by  Focal  Radii.  A  number  of  important 
curves  are  defined  by  imposing  conditions  upon  the  distances  of 
any  point  of  the  locus  from  two  fixed  points,  called  foci. 

(1)  A  point  moves  so  that  the  product  of  its  distances  from  two 
fixed  points  is  constant.  Find  the  equation  of  the  path  of  the  particle. 
Let  the  two  fixed  points  Fi  and  F^,  Fig.  154,  be  taken  on  the  a;-axis 
the  distance  a  each  side  of  the  origin,    Call  the  distances  of  P  from 


§227]  LOCI  387 

the  fixed  points  ri  and  rz.    Then  the  variables  ri  and  r2  in  terms  of  x 
and  y  are: 

ri2  =  y2  +  {x  -  ay 

Hence: 

riVz^  =  [y"  +  {x  -  a)2][i/2  +  (x  +  a)']  (2) 

Calling  the  constant  value  of  riVi  =  c^,   we  have  as  the  Cartesian 
equation  of  the  locus : 

[y'  -^{x  -  a)2][^^  +  (x  +  ay]  =  c'  (3) 


F 


Fig.  154. — The  Lemniscate. 

which  may  be  written: 

(y' +  x^  +  a^y  -  4:a^  x^  =  c*  (4) 

(^2  +  2/^)2  +  2a2a;2  +  2a V'  +  a'  -  ^a^x'-  =  c'  (5) 

(a.2  +  ^2)2  =  2a2(:i;2  -  y^)  +  c*  -  a'  (6) 

If  c  =  a  the  curve  is  called  the  lemniscate,  and  the  Cartesian  equa- 
tion reduces  to : 

(a;2  +  2/2)2  =  2a2(x2  -  y^)  (7) 

For  other  values  of  c  the  curves  are  known  as  the  Cassinian  ovals. 
When  c  <  a  the  curve  consists  of  two  separate  ovals  surrounding  the 
foci,  and  for  c  ^  a  there  is  but  a  single  oval.  The  curves  are  shown 
in  Fig.  157.  These  curves  give  the  form  of  the  equipotential  surfaces 
in  a  field  around  two  positively  or  two  negatively  charged  parallel 
wires.  To  construct  the  curves  proceed  as  follows:  In  Fig.  155 
let  the  circle  have  a  radius  c  and  in  156  let  the  circle  have  the 
diameter  c. 

In  Fig.  155  we  can  use  the  theorem:  ''If  from  a  point  without  a 
circle  a  tangent  and  secant  be  drawn,  the  tangent  is  a  mean  proportional 
to  the  entire  secant  and  the  part  without  the  circle.^*  In  Fig.  156  we  can 
use  the  theorem:  "//  from  the  vertex  of  the  right  angle  of  any  right 
triangle  a  perpendicular  b^  dropped  upor^  the  hypotenusej  then  either 


388        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§227 


leg  of  the  triangle  is  a  mean  proportional  between  the  hypotenuse  and 

the  adjacent  segment." 

Then  in  either  Fig.  155 
or  156,  SPi  X  Spi  =  c2  and 
likewise  SP2  X  Sp2  =  c^, 
etc.  Therefore  SPo,  SPi, 
SP2,  .  .  .  are  the  values 
of  r2  that  correspond  to  ri  = 
Spo,  Spi,  Sp2,  ...,  re- 
spectively, and  the  ovals 
may  be  constructed  by  the 
intersection  of  arcs  described 
about  Fi  and  F2  as  centers, 
using  pairs  of  these  values 
as  radii. 

(2)  Construct  the  curve 
such  that  the  ratio  of  the 
distances  of  any  point  of 
the   curve   from,   two    fixed 


Fig. 


155. — Construction  of  the  Constant 
Products  SPn    X  Spn   =  c2. 


points  is  constant. 

Let  the  two  fixed  points  be  A 
and  B,  Fig.  158;  let  the  constant 
ratio  of  the  distances  of  any 
point  of  the  curve  from  the  two 
fixed  points  be  ri/r2  =  m/n. 

To  find  one  point  of  the  locus, 
draw  circles  from  A  and  B  as 
centers  whose  radii  are  in  the 
ratio  m/n.  Let  these  circles  in- 
tersect at  the  point  P.  At  P 
bisect  the  angle  between  PA  and 
PB  internally  and  externally  by 
the  lines  PM  and  PN  respectively. 
The  line  AB  is  then  divided  at  M 
internally  in  the  ratio  MA/ MB 
=  m/n  and  externally  at  'N  in 
the  ratio  NA/NB  =  m/n,  because 
the  bisectors  of  any  angle  of  a 
triangle  divide  the  base  into 
segments  proportional  to  the  adjacent  sides.  Since  the  external 
and  internal  bisectors  of  any  angle  must  be  at  right  angles  to  each 
other,  PM  is  perpendicular  to  PA''  for  any  position  of  P.     Hence 


Fig.  156. — Construction  of  the  Con- 
stant Products  SPn  X  Spn  =  c2. 


§227] 


LOCI 


389 


the  locus  of  P  is  a  circle,  since  it  is  the  vertex  of  a  right  triangle 
described  on  the  fixed  hypotenuse  MN. 


Fig.   157. — The  Lemniscate  and  the  Cassinian  Ovals. 

If  e  stands  for  the  fixed  ratio  r\/ri  and  if  AB  =  2a,  the  student 
should  show : 

2ae_  r)/lf  =  ^(^-"^ 

1  +  e 
2ae 


MA  = 

NA  = 


MN  = 


1  -  e 

r^e2 


OM  = 
MC  = 


1+e 
2ae 


OC 


a(l  +  e^) 
1  -  e2 


The  equation  of  the  circle  should  then  be  found  referred  to  origin 
0  or  M. 


Fig.  158. — Construction  of  the  Curve  n/r^  =  m/n,  or  the  circle  MPN. 


If  a  large  number  of  circles  be  drawn  for  different  values  of  e,  and 
if  similar  circles  be  described  about  B,  then  these  series  of  circles  are 
known  as  the  dipolar  circles.     See  Fig.  159.     In  physics  it  is  found 


390        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§228 

that  these  circles  are  the  equipotential  lines  about  two  parallel  wires 
perpendicular  to  the  plane  of  the  paper  at  A  and  B  and  carrying 
electricity  of  opposite  sign. 


Fig.   159. — The  Dipolar  Circles,  or  a  Family  of  Circles  made  by  drawing 
n/n  =  e  for  Various  Values  of  e. 

Exercises 

1.  Draw  the  locus  satisfying  the  condition  that  the  ratio  of  the 
distances  of  any  point  from  two  fixed  points  ten  units  apart  is  2/3. 

2.  Draw  the  two  circles  which  divide  a  line  of  length  14  internally 
and  externally  in  the  ratio  3/4, 

228.     The  Cycloid.     The  cycloid  is  the  curve  traced  by  a  point 
on  the  circumference  of  a  circle,  called  the  generating  circle, 

B 


O  D        H       C  A 

Fig.   160. — Definition   of  the   Cycloid. 

which  rolls  without  slipping  on  a  fixed  line  called  the  base.  To 
find  the  equation  of  the  cycloid,  let  OA,  Fig.  160,  be  the  base,P  the 
tracing  point  of  the  generating  circle  in  any  one  position,  and  6  the 
angle  between  the  radius  SP  and  the  line  SH  to  the  point  of  con- 
tact with  the  ba.se.     Since  P  was  at  0  when  the  circle  began  to  roll, 

OH  =  ad 


§229]  LOCI  391 

if  a  be  the  radius  of  the  generating  circle.    Since  x  —  OD  and 
y  =  PD,  we  have: 

x  =  OH  -  SP  sine  =  Si(d  -  sin  d)  (1) 

y  =  HS  -  SPcosB  =  a(l  -  cos  6)  (2) 

These  are  the  parametric  equations  of  the  curve.  For  most 
purposes  these  are  more  useful  than  the  Cartesian  equation. 
It  is  readily  seen  from  the  definition  of  the  curve,  that  the  locus 
consists  of  an  unhmited  number  of  loops  above  the  x-axis,  with 
points  of  contact  with  the  x-axis  at  intervals  of  27ra  (the  circum- 
ference of  the  generating  circle)  and  with  maximum  points  at 
X  =  ira,  37ra,  etc. 

From  the  second  of  the  parametric  equations  we  may  write: 

1  -  cos^  =  ij/a  (3) 

The  expression  (1  —  cos  6)  is  frequently  called  the  versed  sine  of 
$,  and  is  abbreviated  vers  6.    Hence  we  have: 

6  =  vers-i  y  /a  (4) 

Also  from  (3) :  cos  6  =  (a  —  y)  I  a 

Hence:  sin^  =  s/l  -  cos^  6  =  ~V2ay  -  y^      (5) 

whence  substituting  (4)  and  (5)  in  the  first  of  the  parametric  equa- 
tions we  have: 


X  =  avers-i  (y/a)  -  \2ay  -  y^  (6) 

which  is  the  Cartesian  equation  of  the  cycloid,  with  the  origin  0 
at  a  cusp  of  the  curve. 

229.  Graphical  Construction  of  the  Cycloid.  To  construct  the 
cycloid,  Fig.  161,  draw  a  circle  of  radius  1.15  inches  and  divide  the 
circumference  into  thirty-six  equal  parts.  Draw  horizontal  lines 
through  each  point  of  division  exactly  as  in  the  construction  of 
the  sinusoid,  Fig.  59.  Lay  off  uniform  intervals  of  1  /5  inch  each 
on  the  X-axis,  marked  1,  2,  3,  .  .  .  Then  from  the  point  of 
division  of  the  circle  pi  lay  off  the  distance  01  to  the  right. 
From  p2  lay  off  02  to  the  right,  from  ps  lay  off  03  to  the  right, 
etc.  The  points  thus  determined  lie  on  the  cycloid.  The  number 
of  divisions  of  the  circumference  is  of  course  immaterial  except 
that  an  even  number  of  division  is  convenient,  and  except  that 


392        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§230 

the  divisions  laid  off  on  the  base  OA  must  be  the  same  length  as 
the  arcs  laid  off  on  the  circle. 

Note  that  by  the  process  of  construction  above,  the  vertical 
distances  from  OX  to  points  on  the  curve  are  proportional  to 
(1  —  cos  6)  and  that  the  horizontal  distances  from  OF  to  points 
on  the  curve  are  proportional  to  {6  —  sin  d.) 


^^aPjO  12  3  4  5  c  A 

Fig.   161. — Construction  of  the  Cycloid. 

The  analogy  of  the  cycloid  to  the  sine  curve  is  brought  out  by 
Fig.  162.  A  set  of  horizontal  lines  are  drawn  as  before  and  also  a 
sequence  of  semicircles  spaced  at  horizontal  intervals  equal  to 
the  intervals  of  arc  on  the  circle.  The  plane  is  thus  divided 
into  a  large  number  of  small  quadrilaterals  having  two  sides 
straight  and  two  sides  curved.     Starting  at  0  and  sketching  the 


'0  TT  27r 

Fig.   162. — Analogy  of  the  Cycloid  to  the  Sinusoid. 

diagonals  of  successive  cornering  quadrilaterals  the  cycloid  is 
traced.  If,  instead  of  the  sequence  of  circles,  uniformly  spaced 
vertical  straight  lines  had  been  used,  the  sinusoid  would  have  been 
drawn.  The  sinusoid  on  that  account  is  frequently  called  the 
"companion  to  the  cycloid.'^ 

230.  Epicycloids  and  Hypocycloids.     The  curve  traced  by  a 
point  attached  to  the  circumference  of  a  circle  which  rolls  Without 


§230]  LOCI  393 

slipping  on  the  circumference  of  a  fixed  circle  is  called  an  epi- 
cycloid or  a  hypocycloid  according  as  the  rolling  circle  touches 
the  outside  or  inside  of  the  fixed  circle.  If  the  tracing  point 
is  not  on  the  circumference  of  the  rolling  circle  but  on  a  radius 
or  radius  produced,  the  curve  it  describes  is  called  a  trochoid  if 
the  circle  rolls  upon  a  straight  Hne,  or  an  epitrochoid  or  ahjrpo- 
trochoid  if  the  circle  rolls  upon  another  circle.  These  curves  will 
be  discussed  in  the  calculus. 

Exercises 

1.  Construct  a  cycloid  by  dividing  a  generating  circle  of  radius 
1.15  inches  into  twenty -four  equal  arcs  and  dividing  the  base  into 
intervals  3/10  inch  each. 

2.  Compare  the  cycloid  of  length  27r  and  height  1  with  a  semi- 
ellipse  of  length  2ir  and  height  1. 

3.  Write  the  parametric  equations  of  a  cycloid  for  origin  C, 
Fig.  160. 

4.  Write  the  parametric  equations  of  a  cycloid  for  origin  B,  Fig.  160. 

5.  Find  the  coordinates  of  the  points  of  intersections  of  the  cycloid 
with  the  horizontal  line  through  the  center  of  the  generating  circle. 

6.  Show  that  the  top  of  a  rolling  wheel  travels  through  space 
twice  as  fast  as  the  hub  of  the  wheel. 

7.  B}^  experiment  or  otherwise  show  that  the  tangent  to  the  cycloid 
at  any  point  always  passes  through  the  highest  point  of  the  generating 
circle  in  the  instantaneous  position  of  the  circle  pertaining  to  that 
point. 

Exercises  for  Review 

1.  Simplify  the  product: 

{x-2  -  V3)(x-  2  -  i  V3)(x  -  2  +  V3)(a:  -  2  +    Vi3). 

2.  Express  in  the  form  c  cos  (a  —  6)  the  binomial: 

30  cos  a  +  40  sin  a. 

3.  Find  tan  6  by  means  of  the  formula  for  tan  {A  -\-  B),ii  6  = 
tan-i  1/2  +  tan-i  1/3. 

4.  Find  sin  ^  if  0  =  sin"!  1/5  +  sin'^  1/7. 

5.  Find  the  equation  of  a  circle  whose  center  is  the  origin  and 
which  passes  through  the  point  14,  17. 

6.  The  first  of  the  following  tests  was  made  in  1875  with  the 
automatic  air  brake  on  a  train  composed  of  cars  weighing  30,000 


394        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§230 

pounds.  The  second  in  1907  with  the  "LN"  brake  on  a  train 
composed  of  cars  weighing  84,000  pounds.  Find  by  use  of  loga- 
rithmic paper  the  equation  connecting  the  speed  and  the  distance 
run  after  application  of  the  brakes. 

Distance  run  after  application  of  brake  Corresponding  speed 


1875 

1907 

Ofeet 

Ofeet 

57.3  miles 
56.0  miles 

per  hour 

50  feet 

70  feet 

55 . 0  miles 

200  feet 

220  feet 

50.0  miles 

350  feet 

360  feet 

45 . 0  miles 

500  feet 

500  feet 

40 . 0  miles 

820  feet 

770  feet 

25 . 0  miles 

950  feet 

880  feet 

15 . 0  miles 

980  feet 

922  feet 

10.0  mUes 

1,010  feet 

940  feet 

5 . 0  miles 

1,020  feet 

954  feet 

0 . 0  miles 

7.  Discuss  the  curve: 

X 

=  aS 

y 

=  a(l  -  cos  6). 

8.  Graph  on 

polar 

paper: 

P' 

=  a2  cos  26. 

9.  A  fixed  point  located  on  one  leg  of  a  carpenter's  "square" 
traces  a  curve  as  the  square  is  moved,  the  two  arms  of  the  square, 
however,  always  passing  through  two  fixed  points  A  and  B.  Find 
the  equation  of  the  curve. 

10.  Find  the  parametric  equations  of  the  oval  traced  by  a  point 
attached  to  the  connecting  rod  of  a  steam  engine. 

11.  The  length  of  the  shadow  cast  by  a  tower  varies  inversely  as 
the  tangent  of  the  angle  of  elevation  of  the  sun.  Graph  the  length 
of  the  shadow  for  various  elevations  of  the  sun. 

12.  From  your  knowledge  of  the  equations  of  the  straight  line  and 
circle,  graph: 

?/  =  ax  +  Va^  —  x^'  * 

(See  Shearing  Motion,  §37.) 

13.  In  the  same  manner,  sketch: 

y  =  a  -^  X  -\-  Va'  —  x^. 

14.  Graph  the  curve: 

y  =  a/x  +  hx^. 
this  curve  a  minimum  value  for  all  positive  values  of  a  and  6? 


J230] 


LOCI 


395 


16.  Find  by  use  of  logarithmic  paper  the  equations  of  the  curves 
of  Fig.  163.  These  curves  give  the  amounts  in  cents  per  kilowatt- 
hour  that  must  be  added  to  price  of  electric  power  to  meet  fixed 
charges  of  certain  given  annual  amounts  for  various  load  factors. 

16.  The  angle  of  elevation  of  a  mountain  top  seen  from  a  certain 
point  is  29°  4'.     The  angle  of  depression  of  the  image  of  the  mountain 


100 
90 

80 
70 


I 

fa   50 


40 


10 


_             

\\\  : 

u\ 

^"1  \\           1  „___ 

___vvv        

\  \'S      

"tS \-- 

'^^^^—■--"^S^^-- 

i     T 

0.1     0.2     0.3     0.4 


0.5     0.6     0.7     0.8     0.9       1 
Cents  per  K.W.Hour 


11     1-2     1.3     1.4     1.5 


Fig.  163.— Annual  Fixed  Charges  of    $10,   $15  and  $20  of  an  Hydro- 
electric Plant,  Reduced  to  Cents  per  kw-hour  for  Various  Load  Factors. 


top  seen  in  a  lake  230  feet  below  the  observer  is  31°  20'.     Find  the 
height  and  horizontal  distance  of  the  mountain  top,  and  produce  a 
single  formula  for  the  solution  of  the  problem. 
17.  Find  the  points  of  intersection  of  the  curves : 

X2  +  1/2   =  4 


18.  Solve  llOx-4  +  1  =  21x-2. 

19.  Solve  S{x  -  7){x  -  \){x  -  2)  =  (re  +  2){x  -  7){x  +  3). 

20.  Solve  sin  x  cos  x  =  1/4. 

21.  State  the  remainder  theorem  and  illustrate  by  an  example. 


39G        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§230 


22.  Find  the  compound  interest  on  $1000  for  twenty-five  years  at 
5  per  cent.     Show  how  to  solve  by  means  of  progressions. 

23.  The   curve   y^  =  x^   appears   in   which   quadrants?     Inw^hat 
quadrants  is  i/^  =  x^l     Compare  the  curves  x^y"^  =  1  and  x^y^  =  1. 

24.  Which  trigonometric  functions  of  6  increase  as  6  increases  in 
the  first  quadrant?     Which  decrease? 

25.  Given  sin  30°  =  1/2,  cos  45°  =    Vl/2.     Find  the  following: 
sin  150°,  cos  135°,  sin  225°,  cos  300°,  sin  330°,  sin  (-  30°). 

26.  Which  is  greater,  tan  7°  or  sin  7°,  and  why?     Which  is  greater, 
sec  5°  or  esc  5°,  and  why? 

27.  Sketch  the  curves: 

(o)  a;2  +  4x  +  7/2  -  61/  =  12 


(6) 


^2  j^  4^2  -f  62/  =  21. 


28.  From  the  graph  oi  y  =  x''-  obtain  the  graph  of  4?/ 
y  =  4x^ 


x"^  and  of 


m 
10 

/ 

N 

N 

/ 

\ 

8 

/ 

N 

v\ 

/ 

\\ 

t) 

/ 

\ 

1 

^ 

/ 

\  ^ 

/ 

\ 

2 

A 

0 

\ 

/ 

^ 

1 

> 

200        400 


800      1000    m 


Fig.   164. — Trajectory  of  a  German  Army  Bullet  for  a  Range  of    1000 

Meters. 

29.  Given,  cos  d  =  2/5;  find  sin  6,  tan  6  and  cot  d. 

30.  Find  the  equations  of  the  six  straight  lines  determined  by  the 
intersections  of: 

a:2  +  2/2  =  25 
3.2  _  ^2  =  7_ 

31.  In  Fig.  164  the  full  drawn  curve  is  the  trajectory  of  the  pro- 
jectile of  a  German  Army  bullet  for  a  range  of  1000  meters.  The 
dotted  curve  is  the  theoretical  trajectory  that  would  have  been  de- 
scribed by  the  bullet  if  there  had  been  no  air  resistance.     The  dotted 


§230] 


LOCI 


397 


curve  is  a  parabola  (of  second  degree).     Find  its  equation,  taking 
the  necessary  numerical  data  from  the  diagram. 

32.  Find  the  maximum  value  of  p  if  p  =  3  cos  0  —  4  sin  d. 

33.  Find  the  maximum  value  of  y  ii  y  —  ^3  cos  x  —  sin  x,  and 
find  the  value  of  x  for  which  y  is  a  maximum. 

34.  In  Fig.  165  let  ABCO  be  a  square  of  side  a.  Show  that 
for  all  positions  of  ON,  CM  X  AN  =  a^,  and  hence  show  how  to  use 
this  diagram  in  the  construction  of  a  lemniscate. 


Fig.  165.— Construction  of  a  Constant  Product  CM  X  AN  =  AB\ 


CHAPTER  XIII 


THE  CONIC  SECTIONS 


231.  The  Focal  Radii  of  the  Ellipse.  Draw  any  ellipse  with 
major  and  minor  circles  of  radii  a  and  b  respectively,  as  in  Fig. 
166.  Draw  tangents,  IT  and  KK',  to  the  minor  circle  at  the 
extremities  of  the  minor  axes  and  complete  the  rectangle  II'KK'. 
The  points  Fx  and  F^,  in  which  IK  and  I'K'  cut  the  major  axis,  are 


Fig.   166. — Properties  of  the  Ellipse. 


called  the  foci  of  the  ellipse.  From  any  point  on  the  ellipse  draw 
the  focal  radii  PFi  =  ri  and  PF2  =  r2,  as  shown  in  the  figure. 
Represent  the  distance  OFi  or  its  equal  OF 2  by  c.  Then  it  follows 
from  the  triangle  OIFx  that: 

a2  =  b2  +  c2  (1) 

This  is  one  of  the  fundamental  relations  between  the  constants  of 
the  ellipse. 

From  the  triangles  PFiD  and  PF2D  there  follows: 


ri2  =  (c  -  xY  +  2/2 

^2^  =  (c  +  xY  +  y^ 

398 


(2) 
(3) 


§231]  THE  CONIC  SECTIONS  399 

But  the  equation  of  the  ellipse  is 

b 


2/  =  -;   Va 


X' 


or 

y'  =  ->'  -  ^')  (4) 

Substituting  this  value  of  y^  in  (2) 

ri2  =  c2  -  2cx  +  0:2  +  __^(^2  _  ^2)  (5) 

=  c^  -2cx  +  x^  -i-b^ '  x^ 

or  by  (1)  ^  ^,  _  2,^  +  ^.  [1  _  A^]  (6) 

Substituting 


1  _  A'  _  «'  ~  ^' 


a^  a^ 


we  obtain 


Therefore: 


2co:  +  '-V  (7) 


a^ 


r         ca;-|: 


(8) 


r,  =  a  -  f  (9) 

Likewise,  from   (3),    by   exactly   the  same  substitutions,   there 
follows: 

r2  =  a  +  ^  (10) 

From  (9)  and  (10)  by  addition: 

ri+r2  =  2a  (11) 

Hence  in  any  ellipse  the  sum  of  the  focal  radii  is  constant  and  equal 
to  the  major  axis. 

The  converse  of  this  theorem,  namely,  if  the  sum  of  the  focal 
radii  of  any  locus  is  constant,  the  curve  is  an  ellipse,  can  readily 
be  proved.  It  is  merely  necessary  to  substitute  the  values  of  ri 
and  rz  from  (2)  and  (3)  in  equation  (11),  and  simplify  the  resulting 
equation  in  x  and  y;  or  first  square  (11)  and  then  substitute  ri  and 
ri  from  (2)  and  (3).  There  results  an  equation  of  the  second  degree 
lacking  the  term  xy  and  having  the  terms  containing  ,^2  and  y^  both 


400        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§231 

present  and  with  coefficients  of  like  signs.     By  §77,  such  an  equation 
represents  an  ellipse. 

Hence  the  ellipse  might  have  been  defined  as  the  locus  of 
a  point,  the  sum  of  the  distances  of  which  from  two  fixed  points  is 
constant. 

An  ellipse  can  "be  drawn  by  attaching  a  string  of  length  2a  by 
pins  at  the  points  Fi  and  F2  and  tracing  the  curve  by  a  pencil  so 
guided  that  the  string  is  always  kept  taut.  Or  better,  take  a 
string  of  length  2a  +  2c  and  form  a  loop  enclosing  the  two  pins; 
the  entire  curve  can  then  be  drawn  with  one  sweep  of  the 
pencil. 

The  focal  radii  may  also  be  evaluated  in  terms  of  the  parametric 
or  eccentric  angle  6.  The  student  may  regard  the  following 
demonstration  of  the  truth  of  equation  (11)  as  simpler  than  that 
given  above: 

Since  x  =  a  cos  $,  and  y  =  b  sind 

ri2  =  fe2  sin2  0  +  (c  -  «  cos  ^)2  (12) 

=  62  sin2  ^  +  c2  -  2ac  cos  6  -\-  a""  cos^  d  (13) 

To  put  the  right  side  in  the  form  of  a  perfect  square,  write 
62  =  a2  -  c\     Then: 

ri2  =  a2  sin2  0  -  c^  ^[^2  0  -^  c'-  -  2ac  cos  6  +  a^  cos^  d 
=  a2  -  2ac  cos  6  -\-  c^  cos^  d  (14) 

Whence: 

Ti  =  a  —  c  cos  6  (15) 

Likewise: 

Whence: 


12  =  a  +  c  cos  6  (16) 

^1  +  ^"2  =  2a 


232.  The  Eccentricity.  The  ratio  c  /a  measures,  in  terms  of  a  as 
unit,  the  distance  of  either  focus  from  the  center  of  the  ellipse. 
This  ratio  is  called  the  eccentricity  of  the  ellipse.  In  the  triangle 
IFiO,  the  ratio  c/a  is  the  cosine  of  the  angle  FiOI,  represented  in 
what  follows  by  ^.     Calling  the  eccentricity  e,  we  have: 

e  =  c/a  =  cos  jS  (1) 

The  ellipse  is  made  from  the  major  circle  by  contracting  its  ordi- 


§233]  THE  CONIC  SECTIONS  401 

nates  in  the  ratio  m  =  b /a,  or  by  orthographic  projection  of  the 
circle  through  the  angle  of  projection: 

a  =  cos~^  b  /a 
Hence,  as  companion  to  (1)  we  may  write: 

m  =  bja  =  cos  a  =  sin  /3  (2) 

233.  The  Ratio  Definition  of  the  Ellipse.  In  Fig.  166,  let  the 
tangents  to  the  major  circle  at  I  and  I'  be  drawn.  Draw  a 
perpendicular  to  the  major  axis  produced  at  the  points  cut  by 
these  tangents.  These  two  lines  are  called  the  directrices  of  the 
ellipse. 

We  shall  prove  that  the  ratio  PFi  {PH  (or  PF^  }PH')  is  constant 
for  all  positions  of  P.     From  §231,  equation  (9)  or  (15), 

r\  =  a  —  c  cos  6  (1) 

From  the  figure,  ON  =  a  sec  ION  =  a  sec  /3  (2) 

But: 

Hence: 

But 

Therefore 


cos  ^  =  c/a 
ON  =  aVc  (3) 

PH  =  ON  -  X 
PH  =  a^/c  -  a  cose  (4) 


Hence  from  (1)  and  (4) 


'^     =PF,IPH=    "-^^"^ 


PH  ' '  a^Jc  -acosd 

_  c  a  —  c  cos  0 
~  a  a  —  c  cos  6 
or 

PFi/PH  =  c/a  =  e  =  cos  i3  (5) 

A  similar  proof  holds  for  the  other  focus  and  directrix.     Thus, 

for  any  point  on  the  ellipse  the  distance  to  a  focus  bears  a  fixed 

ratio  to  the  distance  to  the  corresponding  directrix.     From  (5), 

the  ratio  is  seen  to  be  less  than  unity. 

Assuming  the  converse  of  the  above,  the  ellipse  might  have  been 
defined  as  follows:  The  ellipse  is  the  locus  of  a  point  whose  distance 
from  a  fixed  point  {called  the  focus)  is  in  a  constant  ratio  less  than 
unity  to  its  distance  from  a  fixed  line  {called  the  directrix). 

26 


402        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§233 

If,  in  any  ellipse,  c  =  0,  it  follows  that  b  must  equal  a  and  the  el- 
lipse reduces  to  a  circle.  If  c  is  nearly  equal  to  a,  then  from  the 
equation: 

a2  =  62  +  c^ 

it  follows  that  the  semi-minor  axis  b  must  be  very  small.     That  is, 
for  an  eccentricity  nearly  unity  the  ellipse  is  very  slender. 

If  the  sun  be  regarded  as  fixed  in  space,  then  the  orbits  of  the 
planets  are  ellipses,  with  the  sun  at  one  focus.  (Th^'s  is  '' Kepler^ s 
First  Law.'')  The  eccentricity  of  the  earth's  orbit  is  0.017.  The 
orbit  of  Mercury  has  an  eccentricity  of  about  0.2,  which  is  greater 
than  that  of  any  other  planet. 

Exercises 

Find  the  eccentricities  and  the  distance  from  center  to  foci  of  the 
following  ellipses : 

1.  a;V9  +  yV4  =  1.  4.  2y  =  Vl  -  x\ 

2.  y  =  (2/3)  V36  -  x\  5.  9^2  +  IQy^  =  14. 

3.  25a;2  +  4^/^  =  100.  6.  2x^  +  Sy^  =  1. 

Find  the  equation  of  the  ellipse  from  the  following  data: 

7.  e  =  1/2,  a  =  4.     Draw  this  ellipse. 

8.  c  =  4,  a  =  5. 

9.  ri  =  6  -  2x/3,  rz  =  6  +  2a;/3. 

10.  ri  =  5  —  4  cos  d,  rz  =5+4  cos  d. 

Solve  the  following  exercises: 

11.  Find  the  eccentricity  of  the  ellipse  made  by  the  orthographic 
projection  of  the  circle  x^  -\-  y^  =  a^  through  the  angle  60°. 

12.  The  angle  of  projection  of  a  circle  x"-  +  y^  =  a'  by  which  an 
ellipse  is  formed  is  a.  Show  that  the  eccentricity  of  the  ellipse  is 
sin  a. 

13.  A  circular  cylinder  of  radius  5  is  cut  by  a  plane  making  an 
angle  30°  with  the  axis.     Find  the  eccentricity  of  the  elliptic  section. 

14.  If  the  greatest  distance  of  the  earth  from  the  sun  is  92,- 
500,000  miles,  find  its  least  distance.      (Eccentricity  of  earth's  orbit 

=  0.017.) 

15.  In  the  ellipse  x'^/25  +  y'^/lQ  =  1,  find  the  distance  between 
the  two  directrices. 

16.  Write  the  equation  of  the  ellipse  whose  foci  are  (2,0),  (  —  2,  0), 
and  whose  directrices  are  x  =  5  and  x  =  —  5.' 


§234]  THE  CONIC  SECTIONS  403 

17.  Prove  equation  11  §231  by  transposing  one  radical  in: 

V(x  +  c)2  +  2/'  +  V(a;  -c)2  +  y""  =  2a 
squaring,  and  reducing  to  an  identity. 

234.  The  Latus  Rectum.  The  double  ordinate  through  the 
focus  is  called  the  latus  rectum  of  the  eUipse.  The  value  of  the 
semi-latus  rectum  is  readily  formed  from  the  equation 


y  =  (b/a)^a^'-  x2 

by  substituting  c  for  x.     HI  represents  the  corresponding  value 
of  2/, 

I  =  {hla)^V-'c^  =  h^/a  (1) 

since  a^  —  c"^  =  b^.     Hence  the  entire  latus  rectum  is  represented 
by: 

21  =  ^-^  (2) 

a 

Equation  (1)  may  also  be  written: 


I  =  bVl  -  c^ja^ 
=  6Vl-e2  (3) 

In  Fig.  166  the  distances  AF,  AN,  ON,  OB,  OF,  FN  may 
readily  be  expressed  in  terms  of  a  and  e  as  follows  in  equations  (4) 
to  (10).  The  addition  of  the  formulas  (11),  (12),  (13)  brings  into 
a  single  table  all  the  important  formulas  of  the  ellipse. 


AFi  =  a  -  c  =  a(l  -  e) 

(4) 

^  _  AFy        a(l-  e) 
e                e 

(5) 

ON  =  a  sec  /3  =  ^ 

(6) 

e  =  cos  j8 

(7) 

OB  =  b  =  a  sin  jS  =  a  Vl  -  e2 

(8) 

OFi  =  c  =  ae 

(9) 

FiN  =  OiV  -  c  =  a(l  -  e2)  /e 

(10) 

1  =  bVa  =  a(l  -  e2) 

(11) 

fi  =  a  —  ex  =  a  -  X  cos  jS 

(12) 

r2  =  a  +  ex  =  a  +  X  cos  jS 

(13) 

404        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§235 

Exercises 

1.  Find  the  value  in  miles  of  OF  for  the  case  of  the  earth's  orbit. 

2.  Find  the  value  of  ^  for  the  earth's  orbit.  (Use  the  S  functions 
of  the  logarithmic  table.) 

3.  In  the  ellipse  y  =  (2/3)  VsG  -  x''  find  the  length  of  the  latus 
rectum  and  the  value  of  e. 

4.  The  eccentricity  of  an  ellipse  is  3/5  and  the  latus  rectum  is  9 
units.     Find  the  equation  of  the  ellipse. 

6.  In  (a)  x^  +  4i/2  =  4  and  (&)  20:2  +  3^2  ^  q  g^d  the  latus 
rectum,  the  eccentricity  and  the  distances  ON  and  AF. 

6.  Determine  the  eccentricities  of  the  ellipses, 

(a)  7/2  =  4a:  -  (l/2)x-^  (6)  y^  =  4x  -  2xK 

7.  Find  the  equation  of  an  ellipse  whose  minor  axis  is  10  units 
and  in  which  the  distance  between  the  foci  is  10. 

8.  Find  the  equation  of  an  ellipse  whose  latus  rectum  is  2  units  and 
minor  axis  is  2. 

9.  The  distance  from  the  focus  to  the  directrix  is  16  units.  An 
ellipse  divides  the  distance  between  focus  and  directrix  externally 
and  internally  in  the  ratio  3/5.     Find  the  equation  of  the  ellipse. 

10.  The  axes  of  an  ellipse  are  known.  Show  how  to  locate 
the  foci. 

11.  In  an  ellipse  a  =  25  feet,  e  =  0.96.     What  are  the  values  of 
c  and  6? 

12.  For  a  certain  comet  (Tempel's)  the  semi-major  axis  of  the 
elliptic  orbit  is  3.5,  and  c  =  1.4  on  a  certain  scale.  For  another 
comet  (Enke's)  a  =  2.2,  e  =  0.85.  Sketch  the  curves,  taking 
3  cm.  or  1  inch  as  unit  of  measure. 

13.  If  I  =  7.2,  e  =  0.6,  find  c,  a,  b. 

14.  An  ellipse,  with  center  at  the  origin  and  major  axis  coinciding 
with  the  X-axis,  passes  through  the  points  (10,  5)  (6,  13).  Find 
the  axes  of  the  ellipse. 

235.  Focal  Radii  of  the  Hjrperbola.  Construct  a  hyperbola  from 
auxiliary  circles  of  radii  a  and  b,  then  the  transverse  axis  of  the 
hyperbola  is  2a  and  the  conjugate  axis  is  26.  Unlike  the  case  of 
the  ellipse,  b  may  be  either  greater  or  less  than  a.  As  previously 
explained,  the  asymptotes  are  the  extensions  of  the  diagonals  of  the 
rectangles  BTAO,  BT'A'O.  From  the  points  I,  I',  in  which  the 
asymptotes  cut  the  a-circle,  draw  tangents  to  the  a-circle.  The 
points  Fi,  F2  in  which  the  tangents  cut  the  axis  of  the  hyper- 
bola are  called  the  foci.     See  Fig.  167. 


§235]  THE  CONIC  SECTIONS  405 

The  distance  OFi  or  OF2  is  represented  by  the  letter  c.  Then, 
since  the  triangles  FJO  and  OAT  are  equal,  FJ  must  equal  6,  so 
that  we  have  the  fundamental  relation  between  the  constants  of 
the  hyperbola: 

a2  4-  b2  =  c2  (1) 


\J 

H 

y 

5 

N\ 

r' 

^y^ 

tV. 

r 

i" 

^ 

\. 

^ 

\ 

N 

T     /  / 

Fy 

y 

V 

A 

1 

JF^ 

D 

/V 

(2) 

e'^/ 

< 

Fig.  167. — Properties  of  the  Hyperbola. 

From  any  point  on  either  branch  of  the  hyperbola  draw  the  focal 
radii  PFi  and  PF2,  represented  by  ri  and  r^  respectively.  Then 
from  the  figure: 

ri2  =  (x  -  cY  +  y^  (2) 

But  from  the  equation  of  the  hyperbola: 

if  =  (6Va')(a;2  -  a^)  (3) 
hence: 

ri2  =  {x  -  c)2  +  h\x''  -  a^)  la''  (4) 

=  {a^x^  -  2a^cx  +  a^c^  +  62^2  _  ^2^2)  ^^2  (5) 

=  (c2ar2  -  2a^cx  +  a^)  /a2  (6) 

=  (ex  -  a^Yla'  (7) 

Hence:      ri  =  {cla)x  —  a  (8) 

In  like  manner  it  may  be  shown  that 

^2  =  (c/a).T  +  a  (9) 


406        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§236 

Hence  from  (8)  and  (9)  it  follows: 

r2  -  ri  =  2a  (lO) 

Hence  in  any  hyperbola,  the  difference  between  the  distances  of  any 

point  on  it  from  the  foci  is  constant  and  equal  to  the  transverse  axis. 

The  above  relation  may  be  derived  in  terms  of  the  parametric 

angle  6.     Thus,  since  in  any  hyperbola  x  =  a  sec  6  and  y  =b  tan^, 

ri2  =  bHsLii'^d  +  {a  seed  -  c)^ 

=  62  tan2  0  +  a2  ^^^2  q  _  2ac  sec  6  -\-  c'^ 

To  put  the  right-hand  side  in  the  form  of  a  perfect  square,  write 


62  =  c2  -  a\ 

Then 

ri2  =  c2  sec2  e  —  2ac  sec  6  +  a^ 

Therefore: 

Ti  =  c  sec  6  —  a. 

and: 

Ti  =  c  sec  ^  +  a 

(11) 

(12) 

236.  The  Ratio  Definition  of  the  Hyperbola.  Through  the 
points  of  intersection  of  the  a-circle  with  the  asymptotes,  draw 
IK,  TK'.  These  lines  are  called  the  directrices  of  the  hyperbola. 
It  will  now  be  proved  that  the  ratio  of  the  distance  of  any  point  of 
the  hyperbola  from  a  focus  to  its  distance  from  the  corresponding 
directrix  is  constant.     Adopt  the  notation: 

cja  =  secjS  =  e  (1) 

Then  from  the  figure: 

PFi  IPH  =  ri  lix  -  ON)  =  r,  /{a  seed  -  a  cos  ^)  (2) 

Substituting  ri  from  (11)  above: 

PFi  IPH  =  (c  sec  d  -  a)  /{a  see  6  -  a  cos  ^)  (3) 

=  (a  sec  j8  sec  6  —  a)  /{a  sec  6  —  a  cos  13)  (4) 

see  ^seeS  -1 

=  o B^  =  sec/3  =  e  =  c  a  (5) 

sec  p  —  cos  d  '  ^  ^ 

which  proves  the  theorem.  The  constant  ratio  e  is  called  the 
eccentricity  of  the  hyperbola,  and,  as  shown  by  (5),  is  always  greater 
than  unity. 

Assuming  the  converse  of  the  above,  it  is  obvious  that  the  hyper- 
bola might  have  been  defined  as  follows:  The  hyperbola  is  the 
locus  of  a  point  whose  distance  from  a  fixed  point  {called  the  focus) 
is  in  a  constant  ratio  greater  than  unity  to  its  distance  from  a  fixed 
line  (called  the  directrix). 


§237]  THE  CONIC  SECTIONS  407 

237.  The  Latus  Rectum.  The  double  ordinate  through  the  focus 
is  called  the  latus  rectum  of  the  hyperbola.  The  value  of  the 
semi-latus  rectum  is  readily  found  from  the  equation: 


y  =  {b/a)  V 


by  substituting  c  for  x.    If  I  represents  the  corresponding  value  of 
y\  

I  =  ib/a)  ylc^  -  02  =  h^a  (1) 

Hence  the  entire  latus  rectum  is  represented  by: 

21  =  2b  Va  (2) 

Equation  (1)  may  also  be  written: 

l  =  byle^-l  "(3) 

In  Fig.  167  the  distances  AFi,  AN,  ON,  OB,  OFi,  FiN  may 
readily  be  expressed  in  terms  of  a  and  e,  as  follows  in  equations 
(4)  to  (8).  Collecting  in  a  single  table  the  other  important  for- 
mulas for  the  hyperbola,  we  have: 

AFi  =  c  -  a  =  a(e  -  1)  (4) 

AN  =  AFJe  =  a(e  -  l)/e  (5) 

ON  =  acos/3  =  a/e  (6) 

e  =  sec  /3  (7) 

0B=b  =  atanj(3  =  a  Ve^  -  1  (8) 

OFi  =  c  =  ae 

FiN  =  c  -  OiV  =  ae  -  a/e  =  a(e2  -  1) /e  (9) 

1  =  bVa  =  b  Ve2  -  1  =  a(e2  -  1)  (10) 

Ti  =  ex  —  a  =  X  sec  ^  —  a.  (11) 

r2  =  ex  -f-  a  =  X  sec  )S  +  a  (12) 

The  important  properties  of  the  hyperbola  are  quite  similar 
to  those  of  the  ellipse.  It  is  a  good  plan  to  compare  them  in 
parallel  columns. 


408        ELEMENTARY  MATHEMATICAL  AN.iLYSlS       [§237 


Ellipse 


1.  Definition  of   Foci  and   Focal 
Radii 


Hyperbola 


1.  Definition  of  Foci  and  Focal 
Radii 

2.  a2  =  5^  +  c2  !  2.  a2  +  6^  =  c^ 

3.  ri  +  r-i  =  2a  3.  rz  —  ri  =  2a 

c  c 

4.  Eccentricity,  e  =  ~-   —  cos  jS  4.  Eccentricity,  e  =  —  =  sec  /3 

5.  Definition  of  Directrices  5.  Definition  of  Directrices 

PFi  PFx 

6.  The  Ratio  Property,  pv>   =  e      6.  The  Ratio  Property,  pjr   =  e 

262  I  262 

7.  The  Latus  Rectum  =  —  7.  The  Latus  Rectum  =  — 

a  I  a 

Exercises 

1.  Find  the  eccentricity  and  axes  of  x^/4l  —  y^/lQ  =  1. 

2.  Find  the  eccentricity  and  latus  rectum  of  the  hyperbola  con- 
jugate to  the  hyperbola  of  the  preceeding  exercise. 

3.  A  hyperbola  has  a  transverse  axis  equal  to  14  units  and  its 
asymptotes  make  an  angle  of  30°  with  he  x-axis.  Find  the  equation 
of  the  hyperbola. 

4.  Find  the  latus  rectum  and  locate  the  foci  and  asymptotes  of 
4a;2  -  36^/2  =  144. 

5.  Locate  the  directrices  of  the  hyperbola  of  the  preceding  exercise. 

6.  In  Fig.  167  show  that  ra  =  GK'  and  ri  =  GI  and  hence  that 
Ti  —  ri  =  IK'  or  2a. 

7.  Find  the  equation  of  the  hyperbola  having  latus  rectum  4/3 
and  a  =  26. 

8.  The  eccentricity  of  a  hyperbola  is  3/2  and  its  directrices  are 
the  lines  x  =  2  and  x  =  —  2.  Write  the  equation  and  draw  the 
curve  with  its  asymptotes,  a-circle,  6-circle,  and  foci. 

9.  Find  the  eccentricity  and  axes  of  Sx-  —  by"^  =  —  45. 

10.  Find  the  eccentricity  of  the  rectangular  hyperbola. 

11.  Describe  the  shape  of  9  hyperbola  whose  eccentricity  is  nearly 
unity.  Describe  the  form  of  a  hyperbola  if  the  eccentricity  is  very 
large. 

12.  Describe  the  hyperbola  if  6/a  =  2,  but  a  very  small. 

13.  Write  the  equation  of  the  hyperbola  if  (1)  c  =  5,  a  =  3;  (2) 
c  =  25,  a  =  24;  (3)  c  =  17,  6  =8. 

14.  Describe  the  locus: 

(:,  +  1)2/7  _  (y  _  3)2/5  =  1. 


§238] 


THE  CONIC  SECTIONS 


409 


15.  Find  the  equation  of  the  hyperbola  whose  center  is  at  the 
origin  and  whose  transverse  axis  coincides  with  the  x-axis  and  which 
passes  through  the  points  (4.5,  —  1),  (6,  8). 

238.  The  Polar  Equation  of  the  Ellipse  and  Hyperbola.     In 

mechanics  and  astronomy  the  polar  equations  of  the  ellipse  and 
hyperbola  are  often  required  with  the  pole  or  origin  at  the  right  focus 
in  the  case  of  the  ellipse  and  at  the  left  focus  in  the  case  of  the  hyper- 
bola.    In  these  positions  the  radius  vector  of  any  point  on  the 


Fig.  168. — Polar  Equation  of  a  Conic. 

curve  will  increase  with  the  vectorial  angle  when  ^  <  180°.  To 
obtain  the  polar  equation  of  the  ellipse  and  hyperbola,  make  use  of 
the  ratio  property  of  the  curves,  namely:  that  the  locus  of  a  point 
whosedistances  from  a  fixed  point  (called  the  focus)  is  in  a  constant 
ratio  e  to  its  distances  from  a  fixed  line  (called  the  directrix),  is  an 
ellipse  if  e  <  1  or  a  hyperbola  if  e  >  1.  In  Fig.  168  letiF'  be  the 
fixed  point  or  focus,  IK  the  fixed  line  or  directrix,  P  the  moving 
point,  arid  FL  =  /  the  semi-latus  rectum.  Then  the  problem  is 
to  find  the  polar  equation  from  the  equation 

PF 

pS  =  e  (1) 


410        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§238 

If  e  is  left  unrestricted  in  value,  the  work  and  the  result  will  apply 
equally  well  either  to  the  ellipse  or  to  the  hyperbola. 

When  the  point  P  occupies  the  position  L,  Fig.  168,  we  have 
PF  ==  I  and  PH  =  FN,  whence  from  (1) 

FN  =  \  (2) 

Take  the  origin  of  polar  coordinates  at  F,  and  also  take  FP  =  p 
and  the  angle  AFP  =  6.     Then: 

PH  =  FN  -  FD  (3) 

FD  =  p  cos  e  (4) 
Hence  from  (2),  (3)  and  (4) 

PH  =  ^-  -  p  cos  e  (5) 

Substituting  these  values  of  FP  and  PH  in  (1),  clearing  of  frac- 
tions and  solving  for  p,  we  obtain 

"  =  l  +  ecosfl  (**' 

which  is  the  equation  required. 

When  e  <  1,  (6)  is  the  equation  of  an  ellipse  with  pole  at  the 
right-hand  focus.  When  e  >  1,  (6)  is  the  equation  of  a  hyperbola 
with  the  pole  at  the  left  focus;  in  both  cases  the  origin  has  been  so 
selected  that  p  increases  as  0  increases. 

Note:  Calling  FN  (Fig.  168)  =  n,  equation  (1)  above  may  be 
written  in  rectangular  coordinates : 


or, 

a;2  -}-  2/2  =  eHn  -  x)^  (8) 

which  may  be  reduced  to  the  form: 

r  ^  1  -  eV     +  1  -  e2   -   (1  -  e2)2  ^^^ 

By  §§77  and  87  this  represents  an  ellipse  if  e  <  1  or  a  hyper- 
bola if  e  >  1.  Thus  starting  with  the  ratio  definition  (7)  we  have 
proved  that  the  curve  is  an  ellipse  or  a  hyperbola;  that  is,  we  have 
proved  the  statements  in  italics  at  bottom  of  pp.  401  and  406. 


§239]  THE  CONIC  SECTIONS  411 

Exercises 

1.  Graph  on  polar  paper,  form   MS,   the   curve  p  =  j~j- . 

for  e  =  2;  also  for  e  =  1/2,  also  for  e  =  1. 

It  will  be  sufficient  in  graphing  to  use  6  =  0°,  30°,  60°,  90°,  120°, 
150°,  180°,  210°,  .    .    .,  360°. 

2.  Write  the  polar  equation  of  an  ellipse  whose  semi-latus  rectum 
is  6  feet  and  whose  eccentricity  is  1/3. 

3.  Write  the  polar  equation  of  an  ellipse  whose  semi-axes  are  5 
and  3. 

4.  Discuss  equation  (6)  for  the  case  6  =  0. 

6.  Write  the  polar  equation  of  a  hyperbola  if  the  eccentricity  be 
'\2  and  the  distance  from  focus  to  vertex  be  4. 
6.  Write  the  polar  equations  of  the  asymptotes  of 


6 


p 
See  §87. 


4  +  5  cos  0 


9  9 

7.  Compare  the  curves  p  =  ~,  j-  k a  ^^^  P  ^ 


4  +  5  cos  0  '^       4  —  5  cos  0 

I 

8.  Discuss  the  equation  p  =  ij — , tt r,  m  which  a  is  a 

^  '^       1  +  e  cos  {d  —  ay 

constant. 

239.  Ratio  Definition  of  the  Parabola.  Among  the  curves  of  the 
parabolic  type  previously  discussed,  the  one  whose  equation  is  of 
the  second  degree  is  of  paramount  importance.  On  that  account 
when  the  term  parabola  is  used  without  qualification,  it  is  under- 
stood that  the  curve  is  the  parabola  of  the  second  order,  whose 
equation  may  be  written,  y"^  =  ax  or  x^  =  ay. 

The  locus  of  a  point  whose  distance  from  a  fixed  point  is  always 
equal  to  its  distance  from  a  fixed  line  is  a  parabola.  In  Fig. 
169,  let  F  be  the  fixed  point  and  HK  the  fixed  line.  Take  the  ori- 
gin at  A  half  way  between  F  and  HK.  Let  P  be  any  point  satis- 
fying the  conditionPi^  =  PH.  Call  OD  =  x,  PD  =  y,  and  represent 
the  given  distance  FK  by  2p.     Then,  from  the  right  triangle  PFD: 

PF^  =  2/'  +  FD^  (1) 

=  y^  -\-  {x  -  OFY 
=  y''-\-{x-  vY 


412        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§240 
Since  PF  by  definition  equals  PH  ov  x  -\-  y,  we  have: 

{x  +  vY  =  y'  +  {x-  vY  (2) 

whence : 

y2  =  4px  (3) 

which  is  the  equation  of  the  parabola  in  terms  of  the  focal  distance, 
OF  or  V. 

The  double  ordinate  through  F  is  called  the  latus  rectum. 

The  semi-latus  rectum  can  be  computed  at  once  from  (3)  by 
placing  X  =  p,  whence: 

1  =  2p  (4) 

where  I  is  the  semi-latus  rectum.  Hence  the  entire  latus  rectum  is 
4p,  or  the  coefficient  of  x  in  equation  (3). 


H 

X'     .^^>^^^  K 

\ 

0 

Vlx^/ 

\             ^ 

^T 

A 

\f                 l 

>       N 

Fig.   169. — Properties  of  the  Parabola  y"^  =  4par. 


In  Fig.  169,  the  quadrilateral  FLIK  is  a  square  since  FL  and 
FK  are  each  equal  to  2^. 

240.  Polar  Equation  of  the  Parabola.  In  accordance  with  the 
ratio  definition  of  the  parabola,  its  polar  equation  is  found  at 
once  from  equation  (6),  §238,  by  putting  e  =  1.  Hence  the  polar 
equation  of  the  parabola  is 

1 

^~  l  +  cos  Q 


(1) 


§241]  THE  CONIC  SECTIONS  413 

For  this  equation  we  may  make  the  following  table  of  values: 
e      \   D 


0° 

1/2 

90° 

I 

180° 

<x> 

270° 

I 

This  shows  that  the  parabola  has  the  position  shown  in  Fig.  168. 
This  is  the  form  in  which  the  polar  equation  of  the  parabola  is 
used  in  mechanics  and  astronomy. 

241.  The  Conies.  It  is  now  obvious  that  a  single  definition 
can  be  given  that  will  include  the  ellipse,  hyperbola  and  parabola. 
These  curves  taken  together  are  Called  the  conies.  The  definition 
may  be  worded:  A  conic  is  the  locus  of  a  point  whose  distances 
from  a  fixed  point  (called  the  focus)  and  a  fixed  line  {called  the 
directrix)  are  in  a  constant  ratio.  The  unity  between  the  three 
curves  was  shown  by  their  equation  in  polar  coordinates.  Moving 
the  ellipse  so  that  its  left  vertex  passes  through  the  origin,  as  in 
§76,  and  writing  the  hyperbola  with  the  origin  at  the  right  ver- 
tex (so  that  both  curves  pass  through  the  origin  in  a  comparable 
manner),  we  may  compare  each  with  the  parabola  as  follows: 


The  ellipse: 

y^  =  2lx  -  {b^la^)x^ 

(1) 

The  parabola: 

?/2  =  2lx 

(2) 

The  hyperbola: 

2/2  =  2lx  +  {b^la^)x^ 

(3) 

In  these  equations  I  stands  for  the  semi-latus  rectum  of  each 
of  the  curves.     These  equations  may  also  be  written: 

2/2  =  2lx  -  {lja)x^  (4) 

2/2  =  2lx  (5) 

2/2  =  2lx  +  {l/a)x^  (6) 

whence  it  is  seen  that  if  I  be  kept  constant  while  a  be  increased 
without  limit,  the  ellipse  and  hyperbola  each  approach  the  para- 
bola as  near  as  we  please.  Only  for  large  values  of  x,  if  a  be  large, 
is  there  a  material  difference  in  the  shapes  of  the  curves. 


414        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§241 

Exercises 

1.  Write  the  equation  of  the  circle  in  the  form  (1)  above. 

2.  Write  the  equation  of  the  equilateral  hyperbola  in  the  form  (3) 
above. 


Y 

1/ 

\ 

45° 

/ 

/ 

^      / 

0 

X 

\  / 

\ 
\ 

\ 

1 

• 

Fig.   170. — A  Hyperbola  Translated  at  an  Angle  of  45°  to  OX. 


Y 

/0^ 

o 

*""'"""■*— .c 

A  w  Ci  10  Co 

Fig.   171.— a  Parabola  Translated  Fig.   172.— Bridge  Truss  in  Form  of 
at  an  Angle  of  60°  to  OX.  Circular  Segment. 


3.  Describe  the  curve: 


where  a  is  a  constant. 


1  +  cos  {Q  —  ot) 


§242] 


THE  CONIC  SECTIONS 


415 


4.  In  Fig.  170  translate  the  curve  xy  =  1  by  suitable  change 
in  the  equation  to  the  position  shown  by  the  dotted  curve,  if  the 
translation  of  each  point  is  unity. 

5.  In  Fig.  171  translate  the  curve  y^  =  4px  by  suitable  change  in 
the  equation  to  the  position  shown  by  the  dotted  curve,  if  the 
distance  each  point  is  moved  be  3p. 

6.  A  bridge  truss  has  the  form  of  a  circular  segment,  as  shown  in 
Fig.  172.  If  the  total  span  be  80  yards  and  the  altitude  BS 
be  20  yards,  find  the  ordinates  CiDi,  C2D2  erected  at  uniform  intervals 
of  10  yards  along  the  chord  AAi. 


X                     1 

3       6         9       12 

^^"^ 

1  r^ 

^1                              B 
Y 

Co     Ci     C      A 

Fig.   173. — Bridge  Truss  in  the  Form  of  a  Parabolic  Segment. 

7.  A  bridge  truss  has  the  form  of  a  parabolic  segment,  as  shown  in 
Fig.  173.  The  span  AAi  is  24  yards  and  the  altitude  OB  is  10 
yards.  Find  the  length  of  the  ordinates  DC,  DiCi,  .  .  .  erected  at 
uniform  intervals  of  3  yards  along  the  line  AAi. 

242.  *  The  Conies  are  Conic  Sections.  The  curves  now  known 
as  the  conies  were  originally  studied  by  the  Greek  geometers  as  the 
sections  of  a  circular  cone  cut  by  a  plane.  At  first  these  sections 
were  made  by  passing  a  plane  perpendicular  to  one  element  of  a 
right  circular  cone.  If  the  angle  at  the  apex  of  the  cone  was  a 
right  angle,  the  section  was  called  the  section  of  the  right  angled 
cone.  If  the  angle  at  the  apex  of  the  cone  was  less  than  90°,  the 
section  made  by  the  cutting  plane  was  called  the  section  of  the 
acute  angled  cone.  Likewise  a  third  curve  was  named  the  section 
of  the  obtuse  angled  cone.  Thus  the  curves  of  three  different 
types  now  called  the  parabola,  ellipse,  and  hyperbola  were  studied. 
The  present  names  were  not  introduced  until  much  later,  and 
until  it  was  shown  that  the  three  classes  of  curves  could  be  made 
respectively  by  cutting  any  cone:  (1)  by  a  plane  parallel  to  an 
element;  (2)  by  a  plane  cutting  opposite  elements  of  the  same  nappe 


416        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§242 

of  the  cone;  (3)  by  a  plane  cutting  both  nappes  of  the  cone.  The 
two  nappes  of  a  conical  surface,  it  will  be  remembered,  are  the 
two  portions  of  the  surface  separated  by  the  apex. 

In  Fig.  174,  let  the  plane  NDN'D',  called  the  cutting  plane,  cut 
the  lower  nappe  of  a  right  circular  cone  in  the  curve  VPV.  We 
shall  prove  that  this  curve  is  an  ellipse. 

Let  the  plane  VAV  pass  through  the  axis  of  the  cone.  It  is 
then  possible  to  fit  into  the  cone  two  spheres  which  will  be  tangent 
to  the  elements  of  the  cone  and  also  tangent  to  the  cutting  plane. 


Fig.  174. — Section  of  a  Circular  Cone. 

For  it  is  merely  necessary  to  locate  by  plane  geometry  the  circle 
inscribed  in  the  triangle  AW,  and  the  escribed  circle  RF'R',  and 
then  to  rotate  these  circles  about  the  axis  AB  to  describe  the 
required  spheres  while  the  line  AR  describes  the  conical  surface. 

Let  the  points  at  which  the  cutting  plane  touches  the  two 
spheres  be  called  F  and  F'. 

From  any  point  P  on  the  curve  VPV  draw  lines  PF  and  PF' 
to  the  points  F  and  F'.  These  lines  are  tangent  to  the  spheres, 
since  each  lies  in  a  tangent  plane  and  passes  through  the  point  of 
tangency.     Through  P  draw  an  element  of  the  cone  AHPK.    The 


§242]  THE  CONIC  SECTIONS  417 

lines  PH  and  PK  are  also  tangents  to  the  upper  and  lower  spheres 
respectively.  Since  all  tangents  to  the  same  sphere  from  the  same 
external  point  are  equal: 

PF   =  PH 
PF'  =  PK 
Hence: 

PF  +  PF'  =PH  +  PK 
But  PH  +  PK  is  an  element  of  the  ivu^iumSHS'  RKR',  and  hence 
preserves  the  same  value  for  all  positions  of  P.     Hence: 

PF  +  PF'  =  a  constant  sum 
Therefore  the  section  is  an  ellipse  with  foci  F  and  F'. 

Let  the  upper  and  lower  circle  of  tangency  of  the  spheres  and 
conical  surface,  namely  SHS'  and  RKR',  be  produced  until  they 
cut  the  cutting  plane  in  the  straight  lines  ND  and  N'D'.  DPD' 
is  a  perpendicular  at  P  to  the  parallel  lines  ND  and  N'D'.  We 
shall  show  that  the  parallel  lines  ND  and  N'D'  are  the  directrices 
of  the  ellipse. 

Since: 

PF  =  PH 
we  have 

PF/PD  =  PH/PD 
The  two  intersecting  lines  DD'  and  HK  are  cut  by  the  parallel 
planes  DNS  and  D'N'R'.     Hence  we  have  the  proportion: 

PH/PD  =  PK/PD'  =  HK/DD' 
This  last  ratio,  however,  has  the  same  value  for  all  positions  of 
P,  since  HK  is  an  element  of  the  frustum  and  since  DD'  is  the 
fixed  distance  between  the  parallel  lines  ND  and  N'D'. 

Therefore  with  respect  to  the  points  F  and  F'  and  the  lines 
ND  and  N'D'  the  ratio  definition  of  the  ellipse  applies  to  the 
curve  VPV.  It  is  easy  to  show  that  the  ratio  HK/DD  is  less 
than  unity. 

If  the  cutting  plane  be  passed  parallel  to  the  element  AR', 
it  is  easy  to  prove  that  the  curve  of  the  section  satisfies  the 
ratio  definition  of  the  parabola.  In  case  the  cutting  plane  cuts 
both  nappes,  one  of  the  tangent  spheres  lies  above  the  apex  and 
it  is  easy  to  show  that  PK  —  PH  is  constant. 

27 


418        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§243 

243.  Tangent  to  the  Parabola.  Let  us  investigate  the  condition 
that  the  Hne  y  =  mx  +  b  shall  be  tangent  to  the  parabola  y-  = 
4px.  First  find  the  points  of  intersection  of  these  loci  by  solving 
the  two  equations  for  x  and  y: 

y    =  mx  -\-  b  (1) 

?/2  =  ipx  (2) 

as  simultaneous  equations. 
Eliminating  y  by  substituting  the  value  of  y  from  (1)  in  (2) 
m2a;2  +  2mbx  -{- b^  -  "^px  =  0  (3) 

or 

m2a;2  +  2{mb  -  2'p)x  +  6^  =  0  (4) 

Solving  for  x  (see  formula  for  quadratic,  Appendix. 


^  _  _  ^^  -  2P  4.  2\/p2  -  ymb  ,.s 

^  ~  m2        -  "~      m2  ^  ^ 

Therefore  there  are  in  general  two  values  of  x  or  two  points  of 
intersection  of  the  straight  Hne  and  the  parabola.  By  the  defini- 
tion of  a  tangent  to  a  curve  (§  146)  the  line  becomes  a  tan- 
gent to  the  parabola  when  the  two  points  of  intersection  be- 
come a  single  point;  that  is,  when  the  radical  in  (5)  vanishes. 
This  condition  requires  that: 

p2  —  yrnb  =  0 
or: 

b  =  p/m  (6) 

» 
Therefore  when  b  of  equation  (1)  has  this  value,  the  line  touches 

the  parabola  at  but  a  single  point,  or  is  tangent  to  it.     The 

equation  of  the  tangent  is  therefore: 

y  =  mx  +  P  /m  (7) 

This  line  is  tangent  to  the  parabola  y"^  =  4px  for  all  values  of 
m.  Substituting  in  (5)  the  value  of  6  =  p/m,  we  may  find  the 
abscissa  of  the  point  of  tangency: 

Xi=p/m2  (8) 

Substituting  this  value  of  x  in  (7)  the  corresponding  ordinate  of 
this  point  is  found  to  be: 

yi  =  2p/m  (9) 


§244]  THE  CONIC  SECTIONS  419 

244.  Properties  of  the  Parabola.  In  Fig.  169,  F  is  the 
focus,  HK  is  the  directrix,  PT  is  a  tangent  at  any  point  P. 
The  perpendicular  PN  to  the  tangent  at  the  point  of  tangency  is 
called  the  normal  to  the  parabola.  The  projection  DT  of  the 
tangent  PT  on  the  x-axis  is  called  the  sub  tangent  and  the  pro- 
jection DN  of  the  normal  PN  on  the  x-axis  is  called  the  sub- 
normal. The  line  through  any  point  parallel  to  the  axis,  asPP, 
is  known  as  a  diameter  of  the  parabola. 

(a)  The  subtangent  to  the  parabola  at  any  point  is  bisected  by 
the  vertex.  It  is  to  be  proved  that  OT  =  OD  for  all  positions  of  P. 
Now  OD  is  the  abscissa  of  P,  which  has  been  found  to  be  p  /m^. 
From  the  equation  of  the  tangent: 

y  =  mx  -{-  p  /m 
the  intercept  OT  on  the  a;-axis  is  found  by  putting  y  =  0  and 
solving  for  x.     This  yields: 

x-=  —  p  Im"^ 
This  is  numerically  the  same  as  OD,  hence  the  vertex  0  bisects 
DT. 

{b)  The  subnormal  to  the  parabola  at  any  point  is  constant  and 
equal  to  the  semi-latus  rectum. 

The  angle  DPN  has  its  sides  mutually  perpendicular  to  the 
sides  of  the  angle  DTP,  hence  the  angles  are  equal.     Since  the 
tangent  of  the  angle  DTP  =  m,  therefore: 
tangent  DPN  =  m 
From  properties  of  the  right  triangle  PDN: 
DN  =  PD  tangent  DPN 
=  PDm 

=  {2p /m)m  =  2p 
Since  KF  also  equals  2p,  we  have 

KF  =  DN 
(c)  PFTH  is  a  rhombus.     By  hypothesis  PF  =  PH.     To  prove 
the  figure  PFTH  a  rhombus  it  is  merely  necessary  to  show  that 
FT  =  PH. 
Now: 

FT  =  FO  +  OT 

PH  ^DK  =  D0-\-  OK 


420        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§245 

But: 

OD  =  OT  and  OK  =  FO 
therefore: 

FT  =  PH 

and  the  figure  is  a  rhombus. 

It  follows  that  the  two  diagonals  of  the  rhombus  intersect  at 
right  angles  on  the  ?/-axis. 

(d)  The  normal  to  a  parabola  bisects  the  angle  between  the  focal 
radius  and  the  diameter  at  the  point.     We  are  to  show  that: 

Z  NPF  =  Z  NPR 
Since  FPHT  is  a  rhombus: 

Z  FPT  =  /  TPH 
But: 

Z  TPH  =   Z  RPS 

being  vertical  angles.  From  the  two  right  angles  NPT  and  NPS 
subtract  the  equal  angles  last  named.     There  results: 

Z  FPN  =  Z  NPR 

It  is  because  of  this  property  of  the  parabola  that  the  reflectors 
of  locomotive  or  automobile  headlights  are  made  parabolic. 
The  rays  from  a  source  of  light  at  F  are  reflected  in  lines  parallel 
to  the  axis,  so  that,  in  the  theoretical  case,  a  beam  of  light  is  sent 
out  in  parallel  lines,  or  in  a  beam  of  undiminishing  strength. 

245.  To  Draw  a  Parabolic  Arc.  One  of  the  best  ways  of  de- 
scribing a  parabolic  arc  is  by  drawing  a  large  number  of  tangent 
lines  by  the  principle  of  §244  (c).  Since  in  Fig.  169  the  tan- 
gent is  for  all  positions  perpendicular  to  the  focal  line  FH  at 
the  point  where  the  latter  crosses  OY,  it  is  merely  necessary  to 
draw  a  large  number  of  focal  lines,  as  in  Fig.  175,  and  erect 
perpendiculars  to  them  at  the  points  where  they  cross  the  ?/-axis. 

The  equations  of  the  tangent  lines  in  Fig.  175  are  of  the  form: 

y  =  mx  -\-  p  jm  (1) 

in  which  p  is  the  constant  given  by  the  equation  of  the  parabola, 
and  in  which  m  takes  on  in  succession  a  sequence  of  values  appro- 
priate to  the  large  number  of  tangent  hues  of  the  figure.  These 
lines  ^re  s^id  to  constitute  ^  family  of  lines  and  are  said  to  envelop 


>245] 


THE  CONIC  SECTIONS 


421 


the  curve  to  which  they  are  tangent.     The  curve  itself  is  called 
the  envelope  of  the  family  of  lines. 
The  curve  of  the  supporting  surface  of  an  aeroplane  as  well  as 


Fig.  175. — Graphical  Construction  of  a  Parabolic  Arc  "by  Tangents." 

the  curve  of  the  propeller  blades  is  a  parabolic  arc.     The  curve  of 
the  cables  of  a  suspension  bridge  is  also  parabolic. 

Exercises 

1.  Write  the  equation  of  the    parabola  which    the    family    y  = 
mx  +  7/2m  envelops. 


422        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§246 


2.  Draw  an  arc  of  a  parabola  if  p  =  3  inches. 

3.  At  what  point  is  y  =  mx  -\-  S/m  tangent  to  the  parabola 
2/2  =  12x? 

4.  At  what  point  is  y  =  mx  +  11/m  tangent  to  y^  =  44x? 

5.  Draw  the  family  of  lines  y  =  mx  -\-  1/m  for  m  =  0.4,  m  =  0.6, 
m  =  0.8,  m  =  1,  m  =  2,  m  =  4,  m  =  8. 

246.  Tangent  to  the  Circle.  An  equation  of  a  tangent  line  to 
a  circle  can  be  found  as  in  the  case  of  the  parabola  above  by  finding 
the  points  of  intersection  of: 

y  =  mx  -\-h  (1) 

and 

a;2  +  2/2  =  a2  (2) 

and  then  imposing  the  condition  that  the  two  points  of  intersection 
shall  become  a  single  point.     The  value  of  h  that  satisfies  this 


Fig.  176. 


-The  Equation  of  a  Line  of  Given  Slope,  Tangent  to  a  Given 
Circle. 


condition  when  substituted  in  (1)  gives  the  equation  of  the  re- 
quired tangent.  It  is  easier  to  obtain  this  result,  however,  by  the 
following  method.  In  Fig.  176  let  the  straight  Hne  be  drawn 
tangent  to  the  circle  at  T.  Let  the  slope  of  this  line  be  m. 
Then  m  =  tan  ONT  =  tan  a,  if  a  be  the  direction  angle  of  the  tan- 
gent line.  The  intercept  of  the  line  on  the  !/-axis  can  be  expressed 
in  terms  of  a  and  a : 


h  =  a  sec  a  =  aVl  + 


m' 


(3) 


§247]  THE  CONIC  SECTIONS  423 

Hence  the  equation  of  the  tangent  to  the  chcle  is: 
y  =  mx  +  aVl  +  m^ 
The  double  sign  is  written  in  order  to  include  in  a  single  equation 
the  two  tangents  of  given  slope  m,  as  illustrated  in  the  diagram. 

Exercises 

1.  Find  the  equations  of  the  tangents  to  x-  -\-  y"^  =  16  making  an 
angle  of  60°  with  the  x-axis. 

2.  Find  the  equations  of  the  tangents  to  x^  +  y"^  =  25  making  an 
angle  of  45°  with  the  a:-axis. 

3.  Find  the  equation  of  tangents  to  x^  +  y"^  =  25  parallel  to 
y  =  Sx  -2. 

4.  Find  the  equation  of  tangents  to  x^  +  y^  =  16  perpendicular 
to  2/  =  (l/2)a;  +  3. 

6.  Find  the  equations  of  the  tangents  to  (x  —  3)^  +  (y  —  4)2  =  25 
whose  slope  is  3. 

247.  Normal  Equation  of  Straight  Line.  The  normal  equation 
of  the  straight  line  was  obtained  in  polar  coordinates  in  §69. 
The  equation  was  written: 

p  cos  {6  —  a)  =  a  (1) 

In  this  equation  (p,  6)  are  the  polar  coordinates  of  any  point  on 
the  line,  a  is  the  distance  of  the  line  from  the  origin  and  a  is  the 
direction  angle  of  a  perpendicular  to  the  line  from  the  origin. 
(See  Fig.  177.)     Expanding  cos  (6  —  a)  in  (1)  we  obtain: 

p  cos  d  cos  a  +  P  sin  0  sin  a  =  a  (2) 

But  for  any  value  of  p  and  6,  p  cos  d  =  x  and  p  sin  ^  =  y. 
Hence  (2)  may  be  written  in  rectangular  coordinates: 

X  cos  a  +  y  sin  a  =  a  (3) 

This  also  is  called  the  normal  equation  of  the  straight  line. 
If  an  equation  of  any  line  be  given  in  the  form: 

ax  -\-  hy  =  c  (4) 

it  can  readily  be  reduced  to  the  normal  form.  For  dividing  this 
equation  through  by    Va^  +  fe^: 

a  ,  b  c 

,  X  +  — y  =  — (5) 

Va2  4-  b2  Va2  +  b^  Va^  +  b^ 


Now  alyla^  +  6^  and  fe/Va^  +  6^  may  be  regarded  as  the  cosine 


424        ELEMENTARY  MATHEMATICAL  ANALYSLS       [§248 

and  sine,  respectively,  of  an  angle,  for  a  and  b  are  divided  by  a 
number  which  may  be  represented  by  the  hypotenuse  of  a  right 
triangle  of  which  a  and  b  are  legs.  Calling  this  angle  a,  equation 
(5)  may  be  written: 

X  cos  a  -\-  y  sin  a  =  d  (6) 

which  is  of  the  form  (3)  above.  Inasmuch  as  the  right  side  of  the 
equation  in  the  normal  form  represents  the  distance  of  the  line 
from  the  origin,  it  is  best  to  keep  the  right  side  of  the  equation 
positive.  The  value  of  a  and  the  quadrant  in  which  it  hes  is 
then  determined  by  the  signs  of  cos  a  and  sin  a  on  the  left  side  of 
the  equation.     The  angle  a  may  have  any  value  from  0°  to  360°. 

Illustrations: 

(1)  Put  the  equation  3x  —  4y  =  10  in  the  normal  form.  Here 
a2  -f-  62  _  25.     Dividing  by  5  we  obtain: 

(3/5)x  -  (4/5)1/  =  2 
The  distance  of  this  line  from  the  origin  is  2.     The  angle  a  is  the  angle 
whose  cosine  is  3/5  and  whose  sine  is  —  4/5.     Therefore  from  the 
tables : 

a  =  306°  52' 

(2)  Put  the  equation  —  Sx  -\-  4y  =  20  in  the  normal  form. 

Here   cos    a  =  -  3/5,  sin  a  =  4/5,  a  =  4.     Hence   a  =  126°  52'. 

(3)  What  is  the  distance  between  the  lines  (1)  and  (2)?  The  lines 
are  parallel  and  on  opposite  sides  of  the  origin.  Their  distance 
apart  is  therefore  2  +  4  or  6. 

Exercises 

1.  The  shortest  distance  from  the  origin  to  a  line  is  5  and  the  direc- 
tion angle  of  the  perpendicular  from  the  origin  to  the  line  is  30°. 
Write  the  equation  of  the  line. 

2.  The  perpendicular  from  the  origin  upon  a  straight  line  makes 
an  angle  of  135°  with  OX,  and  its  length  is  2V2.  Find  the  equa- 
tion of  the  line. 

3.  Write  the  equation  of  a  straight  line  in  the  normal  form  if 
a  =  60°  and  a  =  V3. 

248.  To  Translate  Any  Point  a  Given  Distance  in  a  Given  Direc- 
tion. To  move  any  point  the  distance  d  to  the  right  we  sub- 
stitute (xi  —  d)  for  X.     To  move  the  point  the  distance  d  in  the 


§249]  THE  CONIC  SECTIONS  425 

y  direction  we  substitute  (t/i  —  d)  for  y.  To  move  any  point 
the  distance  d  in  the  direction  a  we  substitute: 

X  =  Xi  —  d  cos  a 

y  =^  yi  —  d  s'm  a  (1) 

which  must  give  the  desired  position  of  the  new  point.  It  is 
not  necessary  to  use  the  subscript  attached  to  the  new  coordinates 
if  the  distinction  between  the  new  and  old  coordinates  can  be 
kept  in  mind  without  this  device. 

The  circle  x^  +  y^  =  a^  moved  the  distance  d  in  the  direction 
a  becomes: 

{x  —  d  cos  ay  -{-  (y  —  d  sin  a)^  =  «^ 
which  may  be  simphfied  to: 

x^  —  2dx  cos  a  -{-  y^  —  2dy  sin  a  =  a^  —  d"^ 

249.  Distance  of  Any  Point  From  Any  Line.     Let  the  equation 
of  the  line  be  represented  in  the  normal  form: 

X  cos  a  +  ?/  sin  a  =  a  (1) 

and  let  {xi,  yi)  be  any  point  P  in  the  plane.  (See  Fig.  177.) 
If  the  point  (xi,  yi)  is  on  the  same  side  of  the  line  as  the  origin, 
the  point  can  be  •  moved  to  the  line  by  translating  the  point 
the  proper  distance  in  the  a  direction.  Let  the  unknown  amount 
of  the  required  translation  be  represented  by  d.  To  translate 
the  point  P  the  amount  d  in  the  a  direction,  we  must  substitute 
for  Xi  and  ?/i  the  values: 

Xi  =  X  —  d  cos  a  /o\ 

yi  "=  y  —  d  sin  a 
By  hypothesis  the  point  now  lies  on  the  line,  and  therefore  the 
new  coordinates  {x,  y)  of  the  point  must  satisfy  the  equation  of 
the  line.     Hence,  solving  (2)  for  x  and  y  and  substituting  their 
values  in  (1)  we  have: 

{xi  +  d  cos  a)  cos  a  +  (yi  -\-  d  sin  a)  sin  a  =  a  (3) 

Performing  the  multiplications  and  solving  for  the  unknown 
number  d,  we  have: 

d  =  —  (xi  cos  a  +  yi  sin  a  ~  a)  (4) 

This  is  the  distance  of  {xi,  yi)  from  the  line.  Since  this  distance 
would  ordinarily  be  looked  upon  as  a  signless  or  arithmetical 


426        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§249 

number,  the  algebraic  sign  may  be  ignored,  and  only  the  absolute 
value  of  the  expression  be  used.  The  negative  sign  means  that 
the  given  point  lies  on  the  origin  side  of  the  line. 

Equation  (4)  may  be  interpreted  as  follows: 

To  find  the  distance  of  any  point  from  a  given  line,  put  the  equa- 
tion of  the  line  in  the  normal  form,  transpose  all  terms  to  the  left 


Fig.  177. — Normal  Equation  of  a  Line,  and  the  Distance  of  Any  Point 
from  a  Given  Line. 


member  and  substitute  the  coordinates  of  the  given  point  for  x  and 
y.  The  absolute  value  of  the  left  member  is  the  distance  of  P  from 
the  line. 

If  the  given  point  P  and  the  origin  of  coordinates  lie  on  op- 
posite sides  of  the  given  line,  then  the  point  P  (Fig.  177)  must 
be  translated  in  the  direction  (180°  +  a)  to  reach  the  line. 
Hence  the  substitutions  are 

Xi  =  X  -  dco3  (180°  +  a) 
yi  =  y  -  d  sin  (180°  +  a) 


§250]  THE  CONIC  SECTIONS  427 

or, 

Xi  =  X  -^  d  cos  a 

yi  =  y  +  d  sin  a 
Solving  these  for  x  and  y,  substituting  in  the  equation  of  the  line, 
and  solving  for  d  we  obtain: 

d  =  Xi  cos  Q!  +  yi  sin  a  —  a  (6) 

The  absolute  value  is  of  the  same  form  as  before.  Hence  only 
the  one  formula  (4)  is  required.  When  the  result  in  any  problem 
comes  out  negative  it  merely  means  the  given  point  lies  on  the 
origin  side  of  the  hne. 

The  above  facts  may  be  stated  in  an  interesting  form  as  follows: 
Let  any  line  be: 

X  cos  a  -\-  y  sin  a  —  a  =  0 
If  the  coordinates  of  any  point  on  this  line  be  substituted  in  this 
equation,  the  left  member  reduces  to  zero.  If  the  coordinates  of 
any  point  not  on  the  line  be  substituted  for  x  and  y  in  the  equation, 
the  left  member  of  the  equation  does  not  reduce  to  zero,  but 
becomes  negative  if  the  given  point  is  on  the  origin  side  of  the  line 
and  positive  if  the  given  point  is  on  the  non-origin  side  of  the 
line.  The  absolute  value  of  the  left  member  in  each  case 
gives  the  distance  of  the  given  point  from  the  line.  Thus  every 
line  may  be  said  to  have  a  ''positive  side"  and  a  ''negative 
side."    The  "negative  side"  is  the  side  toward  the  origin. 

Exercises 

1.  Find  the  distance  of  the  point  (4,  5)  from  the  line  3x  +  iy  =  10. 

2.  Find  the  distance  from  the  origin  to  the  line  x/S  —  y/4:  =  1. 

3.  Find  the  distance  from  (—3,  —  4)  to: 

12(a;  +  6)  =  5{y  -  2) 

4.  Find  the  distance  from  (3,  4)  to  the  line  a;/3  —  y/4t  =  1. 

6.  Find    the    distance   between    the    parallel    lines    y  =  2x  -\-  3, 
y  =  2x  +  5. 

6.  Find  the  distance  between  y  =  2x  —  3,  y  =  2x  -\-  5. 

7.  Find  the  distance  from  (0,  3)  to  4x  -  3?/  =  12. 

8.  Find  the  distance  from  (0,  1)  to  a;  +  2  -  2?/  =  0. 

250.  Tangent  to  a  Circle  at  a  Given  Point.     The  equation  of 
the  tangent   to  the   circle   obtained    in  §246    is  the  equation 


428        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§251 

of  the  tangent  line  having  a  given  or  required  slope  m.     We 
shall  now  find  the  equation  of  the  line  that  is  tangent  to  the  circle 
at  a  given  point  {xq,  t/o). 
The  line: 

a  =  p  cos  {d  —  a)  (1) 

or  its  equivalent: 

X  cos  a  -\-  y  sin  a  =  a  (2) 

is  tangent  to  the  circle  of  radius  a,  and  the  point  of  tangency  is  at 
the  end  of  the  radius  whose  direction  angle  is  a.  The  point  of 
tangency  is  therefore  {a  cos  a,  a  sin  a).  Hence,  multiplying  (2) 
through  by  a,  we  obtain: 

x{a  cos  a)  +  y{a  sin  a)  =  a^  (3) 

or: 

Xox  +  Yoy  =  a2  (4) 

which  is  the  equation  of  the  line  tangent  at  the  point  {xo,  ?/o)  to 
the  circle  of  radius  a. 

Thus  3x  +  ^y  =  25  is  tangent  to  x^  +  ?/2  =  25  at  the  point 
(3,  4). 


Fig.  178. — Tangent  to  the  Ellipse  at  a  Given  Point. 


251.  Tangent  to  the  Ellipse  at  a  Given  Point.  It  is  easy  to 
draw  the  tangent  to  the  ellipse  at  any  desired  point.  In  Fig.  178, 
let  Po  be  the  point  at  which  a  tangent  is  desired.  Then  draw  the 
major  circle,  and  let  Pi  of  the  circle  be  a  point  on  the  same  ordinate 


§252]  THE  CONIC  SECTIONS  429 

as  Pq.  Draw  a  tangent  to  the  circle  at  Pi  and  let  it  meet  the  x-axis 
at  T.  Then  when  the  circle  is  projected  to  form  the  ellipse,  the 
straight  line  PiT  is  projected  to  make  the  tangent  to  the  ellipse. 
Since  T  when  projected  remains  the  same  point  and  since  Po  is 
the  projection  of  Pi,  the  line  through  Po  and  T  is  the  tangent  to 
the  ellipse  required. 

The  equation  of  the  tangent  Po?"  is  also  readily  found.     The 
equation  oiPiT  is: 

^•^0  +  yy\  =  a"^  (1) 

To  project  this  into  the  line  PqT  it  is  nierely  necessary  to  multiply 
the  ordinates  y  and  2/'oby  6 /a;  that  is,  to  substitute  y  =  ay  /b  and 
2/'o  =  ayo/b.     Whence  (1)  becomes: 

x^-^  a^yoy/b^  =  a^  (2) 

or  dividing  by  a^, 

Xox/a2  +  yoy/b2  =  1  (3) 

which  is  the  tangent  to: 

x2/«'  +  y'/b^  =  1 

at  the  point  (a'o,  yo). 

Exercises 

1.  Find  the  equations  of  the  tangents  to  the  ellipse  whose  semi-axes 
are  4  and  3  at  the  points  for  which  x  =  2. 

2.  Find  the  equations  of  the  tangents  to  x'^/lQ  +  y^/9  =  1  at  the 
ends  of  the  left  latus  rectum. 

3.  Required  the  tangents  to  x^/Q  +  y^/A  =  1  making  an  angle  of 
45°  with  the  a:-axis. 

4.  Find  the  equations  of  the  tangents  to  x^/lOO  +  y^/25  =  1  at 
the  points  where  ^  =  3. 

5.  Find  the  equations  of    the   tangents  to  xV36  +  2/V16  =  1  at 
the  points  where  x  =  y. 

252.  The  Tangent,  Normal,  and  Focal  Radii  of  the  Ellipse.    In 

the  right  triangle  PiOT,  Fig.  178,  the  side  PiO  is  a  mean  propor- 
tional between  the  entire  hypotenuse  OT  and  the  adjacent 
segment  OD,     That  is: 

a2  =  xoOT 
But:  FiT  ^  OT  -  OFi 

E  a^/xg  —  ae 


430        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§253 

Likewise:  F^T  =  OT  +  OF2 

=  a^/xQ  +  ae 
Therefore:  FiT /F2T  =  (a^xo  -  ae)  l(a' /xo  +  ae) 

=  (a  —  ex^i)  /{a  +  exo) 
But   by   §231  this  last  ratio  is  equal  to  ri/r2.     Therefore   we 
may  write:  F,T IF2T  =  P^FijP^i. 

Hence  T,  which  divides  the  base  i^aFi  of  the  triangle  PoF^Fi 
externally  at  T  in  the  ratio  of  the  two  sides  PF2  and  PFi  of  the 
triangle,  lies  on  the  bisector  of  the  external  angle  FiPqQ  of  the 
triangle  F2P0F1.     This  proves  the  important  theorem: 

The  tangent  to  the  ellipse  bisects  the  external  angle  between  the 
focal  radii  at  the  point. 

This  theorem  provides  a  second  method  of  constructing  a 
tangent  at  a  given  point  of  an  ellipse,  often  more  convenient 
than  that  of  §251,  since  the  method  of  §251  often  runs  the 
construction  off  of  the  paper. 

The  normal  PqN,  being  perpendicular  to  the  tangent,  must 
bisect  the  internal  angle  F2P0F1  between  the  focal  radii  F2P0  and 

Since  the  angle  of  reflection  equals  the  angle  of  incidence  for 
light,  sound,  and  other  wave  motions,  a  source  of  light  or  sound  at 
Fi  is  ''brought  to  a  focus"  again  at  F2,  because  of  the  fact  that  the 
normal  to  the  ellipse  bisects  the  angle  between  the  focal  radii. 

253.  Additional  Equations  of  the  Straight  Line.  The  equations 
of  the  straight  line  in  the  slope  form: 

y  =  mx  -\-  b  (1) 

and  in  the  normal  forms: 

p  cos  {d  -  a)  =  a  (2) 

x  cos  a  -^  y  sina  =  a  (3) 

and  the  general  form: 

ax  +  by  +  c  =  0  (4) 

have  already  been  used.  Two  constants  and  only  two  are  neces- 
sary for  each  of  these  equations.  The  constants  in  the  first 
equation  are  m  and  b;  in  the  second  and  third,  a  and  a;  in  the 
fourth  a  [c  and  b  jc,  or  any  two  of  the  ratios  that  result  from  divid- 
ing through  by  one  of  the  coefficients.  Equation  (4)  appears  to 
contain  three  constaiits,  but  it  is  only  the  relative  size  of  these  that 


§253]  THE  CONIC  SECTIONS  431 

determines  the  particular  line  represented  by  the  equation,  since 
the  line  would  remain  the  same  when  the  equation  is  multiplied 
or  divided  through  by  any  constant  (not  zero). 

These  facts  are  usually  summarized  by  the  statement  that  two 
conditions  are  necessary  and  sufficient  to  determine  a  straight 
line.  The  number  of  ways  in  which  these  conditions  may  be  given 
is,  of  course,  unlimited.  Thus  a  straight  line  is  determined  if  we 
say,  for  example,  that  the  line  passes  through  the  vertex  of  an 
angle  and  bisects  that  angle,  or  if  we  say  that  the  line  passes 
through  the  center  of  a  circle  and  is  parallel  to  another  line,  or  if 
we  say  that  the  straight  line  is  tangent  to  two  given  circles,  etc. 
An  important  case  is  that  in  which  the  line  is  determined  by  the 
requirement  that  it  pass  through  a  given  point  in  a  given  direc- 
tion. The  equation  of  the  Hne  adapted  to  this  case  is  readily 
found.  Let  the  given  point  be  {xi,  yi).  The  line  through  the 
origin  with  the  required  slope  is 

y  =  mx 
Translate  this  line  so  that  it  passes  through  {xi,  yi)  and  we  have 
y  -  yi  =  m(x  -  Xi)  (5) 

Another  way  of  obtaining  the  same  result  is:  substitute  the 
coordinates  {xi,  yi)  in  (1) : 

2/1  =  wixi  +  b  (6) 

Subtract  the  members  of  this  from  (1)  above,  so  as  to  eliminate 
b.    There  results: 

y  —  yi  =  m{x  —  xi)  (7) 

This  is  the  required  equation;  the  given  point  is  {xi,  yi)  and  the 
direction  of  the  line  through  that  point  is  given  by  the  slope  m. 
Another  important  case  is  that  in  which  the  straight  line  is 
determined  by  requiring  it  to  pass  through  two  given  points. 
Let  the  second  of  the  given  points  be  {x2,  2/2).  Substitute  these 
coordinates  in  (5) : 

yi  —  yi  =  m{xi  -  xi)  (8) 

To  eliminate  m,  divide  the  members  of  (7)  by  the  members  of  (8) : 

y  ~  y^  =  X  -  xi  .Q. 

2/2  -  yi      X2-  xi 


432        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§254 


or,  as  it  is  usually  written: 

y  -  yi     y2  -  yi 


(10) 


X   —   Xi         X2  —  Xi 

This  is  the  equation  of  a  line  passing  through  two  given  points. 
Since  (10)  may  be  looked  upon  as  a  proportion,  the  equation  may 
be  written  in  a  variety  of  forms. 

254.  The  Circle  Through  Three  Given  Points.  In  general,  the 
equation  of  a  circle  can  be  found  when  three  points  are  given. 
Either  of  the  general  equations  of  the  circle: 

(x  -  hy  +  (?/  -  kY  =  a2  (1) 

or: 

x'^  +  if-^  2gx  +  2/i/  +  c  =  0  (2) 

contains  three  unknown  constants,  so  that  in  general  three 
conditions  may  be  imposed  upon  them.  It  is  best  to  illustrate 
the  general  method  by  a  particular  example.  Le-t  the  three 
given  points  be  (—  1,  3),  (0,  2),  and  (5,  0).  Then  since  the  co- 
ordinates of  these  points  must  satisfy  the  equation  of  the  circle, 
we  obtain  from  (2)  above: 

1  +  9  -    26r  +  6/  +  c  =  0  (3) 

4  +  4/  +  c  =  0  (4) 

25  +10^  +  c  =  0  (5) 

Eliminating  c  from  (3)  and  (4)  and  from  (4)  and  (5),  we  obtain: 
6  -    2sr  +  2/  =  0 
21  -\-10g  -4f  =  0 
Eliminating  /: 


g=  -^h 

whence: 

f=-8h 

and 

c  =  30 

So  the  equation  of  the  circle  is: 

x'-^y' 

-  Ux  -  I7y  +  30 

Exercises 

1.  Find   the  equation  of    the  line  passing  through   (2,  3)  with 
slope  2/3. 

2.  Find  the  equation  of  the  line  passing  through  (2,  3),  (3,  5). 


§255]  THE  CONIC  SECTIONS  433 

3.  Find  the  line  passing  through  (2,  —  1)  making  an  angle  whose 
tangent  is  2  with  the  x-axis. 

4.  Find  the  line  through  (2,  3)  parallel  to  y  =  7a;  +  11. 

6.  A  line   passes   through    (  —  1,  —  3)    and    is  perpendicular  to 
y  —  2x  =  S.     Find  its  equation. 

6.  Find  the  line  passing  through  (  —  2,  3),  (  —  3,  —  1). 

7.  Find  the  equation  of  the  line  which  passes  through  (—  1,  —  3), 
(  -  2,  4). 

8.  Find  the  slope  of  the  line  that  passes  through  ( —  1,  6),  ( —  2,  8). 

9.  Find  the  equation  of  the  line  passing  through  the  left  focus  and 
the  upper  end  of  the  right  latus  rectum  of  a;V25  +  ?/V9  =  1. 

10.  Find  the  equation  of  the  circle  passing  through  (2,8),  (5,7), 
and  (6,  6). 

11.  Find  the  equation  of  the  circle  which  passes  through  (1,  2), 
(  -  2,  3),  and  (  -  1,  -  1). 

12.  Find  the  equation  of  the  parabola  in  the  form  y^  =  4:px  which 
passes  through  the  point  (2,  4). 

255.  Change   from   Polar   to  Rectangular  Coordinates.      The 

relations  between  x,  y  of  the  Cartesian  system  and  p,  6  of 
the  polar  system  have  already  been  explained  and  use  made  of 
them.     The  relations  are  here  brought  together  for  reference: 

X  =  p  cos  d  (1) 

y  =  p  sin  ^  (2) 

By  these  we  may  pass  from  the  Cartesian  equation  of  any  locus 
to  the  equivalent  polar  equation  of  that  locus.  Dividing  (2) 
by  (1)  and  also  squaring  and  adding,  we  obtain: 

e  =  tan-\yjx  (3) 

P  =  \/x2  -Ty^  (4) 

These  may  be  used  to  convert  any  polar  equation  into  the  Cartesian 
equivalent. 

256.  Rotation  of  Any  Locus.  It  has  already  been  explained 
that  any  locus  can  be  rotated  through  an  angle  a  by  substituting 
(^1—  a)  for  d  in  the  polar  equation  of  the  locus.  It  remains  to 
determine  the  substitutions  for  x  and  y  which  will  bring  about 
the  rotation  of  a  locus  in  rectangular  coordinates.  Let  us  consider 
any  point  P  of  a  locus  before  and  after  rotation  through  the  given 
angle    a.     Call   the   coordinates   of   the   point   before    rotation 


434        ELEMENTARY  MATHEMATICAL  ANALYSIS 


^256 


(x,  y)  in  rectangular  coordinates  and  (p,  6)  in  polar  coordinates. 

Then,  from  (1)  and  (2),  §255, 

X  =  p  cos  d  (1) 

y  =  p  sin  e  (2) 

Call    the  coordinates  of  the  point  after  rotation    (xi,  2/1)   and 

{pi,  di),  but  note  that  the  value  of  p  is  unchanged  by  the  rotation. 

Then  for  the  point  P',  Fig.  179,  we  may  write : 


Fig.  179. — Rotation  of  Any      Fig.  180. — Effect  of  Rotation  on  t 
Locus.                                    Forms  x^  -\-  y^,  2xy,  and  x^ 

he  Special 
-2/-. 

Xi  =  p  COS  ^1 
?/i  =  p  sin  ^1 

(3) 
(4) 

Since,  however,  the  rotation  requires  that 

d  =  di-  a 

(5) 

equations  (1)  and  (2)  become: 

a;  =  p  cos  (^1  —  a)  =  p  cos  di  cos  a  +  p  sin  di  sin  a        (6) 

2/  =  p  sin  (^1  —  a)  =  p  sin  61  cos  a  —  p  cos  61  sin  a        (7) 

But,  from  (3)  and  (4),  p  cos  di  and  p  sin  di  are  the  new  values  of 

X  and  y;  hence,  substituting  in  (6)  and    7)  from  (3)  and  (4)  we 

obtain: 

X  =  Xi  cos  a  +  yi  sin  a  (8) 

y  =  yi  cos  a  —  Xi  sin  a  (9) 

Hence  if  the  equation  of  any  locus  is  given  in  rectangular  co- 


§256]  THE  CONIC  SECTIONS  435 

ordinates,  it  is  rotated  through  the  positive  angle  a  by  the  sub- 
stitutions 

X  cos  a  +  y  sin  a  for  X 
y  cos  a  —  xsin  afor  y  (10) 

in  which  it  is  permissible  to  drop  the  subscripts,  if  the  context 
shows  in  each  case  whether  we  are  dealing  with  the  old  x  and  y 
or  with  the  new  x  and  ?/. 

If  the  required  rotation  is  clockwise,  or  negative,  we  must 
replace  a  by  ( —  a)  in  all  of  the  above  equations. 

Whenever  convenient,  tine  equation  of  a  curve  should  be  taken  in 
the  polar  form  if  it  is  required  to  rotate  the  locus. 

Important  Facts  :  The  following  facts  should  be  remembered 
by  the  student: 

(1)  To  rotate  a  curve  through  90°,  change  x  to  y  and  y  to  (—  re). 
This  fact  has  been  noted  in  §68. 

(2)  Rotation  through  any  angle  leaves  the  expression  x^  -\-  y^ 
{or  any  function  of  it)  unchanged.  This  is  obvious  since  the  circle 
x2  +  2/2  =,  fj^2  is  j^Qj^  changed  by  rotation  about  (0,  0). 

(3)  Rotation  through  +  45°  changes  2xy  to  y^  —  x^. 
Rotation  through  —  45°  changes  2xy  to  x"^  —  y^. 

(4)  Rotation  through  +  45°  changes  x"^  —  y"^  to  2xy. 
Rotation  through  —  45°  changes  x^  —  y^  to    —  2xy. 

Statements  (3)  and  (4)  follow  at  once  from  consideration  of  the 
equations 

0^2  _  2/2  =  a2  (1) 

2xy  =  a2  (2) 

7/2  -x^  =  a^  (3) 

-  2xy  =  a2  (4) 

of  the  four  hyperbolas  bearing  corresponding  numbers  (1),  (2), 

(3),  (4)  in  Fig.    180.     The  proper  change  in  any  case  can  be 

remembered  by  thinking  of  the  four  hyperbolas  of  this  figure. 

(5)  The  degree  of  an  equation  of  a  locus  cannot  be  changed  by 
a  rotation.  This  follows  at  once  from  the  fact  that  the  equations 
of  transformation  (8)  and  (9)  are  linear. 

Exercises 

In  order  to  shorten  the  work,  use  statements  (1)  to  (4)  whenever 
possible. 


436        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§257 

1.  Turn  the  locus  x''-  —  y^  =  4:  through  45°. 

2.  Turn  x^  -\-  y"^  =  a"^  through  79°.     Turn  4xy  =  1  through  45°. 

3.  Turn  x  cos  a  -{-  y  sin  a  =  a  through  an  angle  /3.  (Since  this  locus 
is  well  known  in  the  polar  form,  transformation  formulas  (6)  and  (7) 
above  may  be  avoided.) 

4.  Rotate  x"^  -  y"^  =  I  through  90°. 

5.  Rotate  x"^  -  y"^  =  a^  through  -  45°. 

6.  Change  the  equation  (a:  —  o)^  +  (^  —  6)^  =  r'^  to  the  polar 
form. 

7.  Change  p  cos  2d  =  2a,  one  of  a  class  of  curves  known  as  Cote's 
spirals,  to  the  Cartesian  form. 

8.  Write  the  equation  of  the  lemniscate  in  the  polar  form. 

9.  Show  that  p^  —  2ppi  cos  (6  —  di)  +  pi^  =  a^  is  the  polar  equa- 
tion of  a  circle  with  center  at  (pi,  di)  and  of  radius  a. 

10.  Write  the  Cartesian  equation  of  the  locus  p^  =  16  sin  2d. 

11.  Turn  p2  =  8  sin  26  through  an  angle  of  45°. 

12.  Rotate  x^  -  2y^  =  1  through  90°. 

13.  Rotate  (x^  +  ?/^)^^  +  (x^  -  y'^Y^  =  1  through  45°. 

14.  Rotate  log  (x^  +  y^)  =  tan  (x^  -  y"")  through  45°. 

257.  EUipse  with  Major  Axis   at  45°  to  the  OX  Axis.    The 

ellipse  frequently  arises  in  applied  science  as  the  resultant  of  the 
projection  of  the  motion  of  two  points  moving  uniformly  on  two 
circles,  as  has  already  been  explained  in  §186.  Thus  the 
parametric  equations: 

X  =  a  cos  t  (1) 

y  =  h  sin  t  (2) 

define  an  ellipse  which  may  be  considered  the  resultant  of  two 
S.H.M.  in  quadrature.     We  shall  prove  that  the  equations: 

a;  =  a  cos  I  (3) 

y  =  a  sin  {t  -}-  a)  (4) 

define  an  ellipse,  with  major  axis  making  an  angle  of  45°  with  OX. 
The  graph  is  readily  constructed  as  in  Fig.  181.  The  Car- 
tesian equation  of  the  curve  is  found  by  eliminating  t  between 
(3)  and  (4).  Expanding  the  sin  {t  +  a)  in  (4)  and  substituting 
from  (3)  we  obtain: 

y  =  X  sin  a-\-  v  a^  —  x^  cos  a  (5) 

Transposing  and  squaring: 

x~  —  2xy  sin  a  -\-  y'^  —  a^  cos^  a  (6) 


§258] 


THE  CONIC  SECTIONS 


437 


By  §256  rotate  the  curve  through  an  angle  of  (—  45°.)  We 
know  that  (x"^  +  y^)  is  unchanged  and  that  2xy  is  to  be  replaced 
by  (a;2  —  y"^).     Therefore  (6)  becomes: 

x2(l  —  sin  a)  -\-  y'^{l  +  sin  a)  =  a^  cos^  a  (7) 


/ 

/^ 

Y 
B 

V 

^ 

i^^^ 

^ 

li'^     N^ 

/ 

^' 

\     ^^ 

X' 

/ 

y' 

■^    ■ 

/  y^"    \x 

L 

.y 

O 

/ 

A 

T  ^ 

■^' 

/ 

/ 

y 

/ 

^, 

^ 

y' 

Fig.   181. — The  Ellipse  x  =  a  cos  t,  y  =  a  sin  {t  +  a). 

Replacing  cos^  a  by   1  —  sin^  a,  and  dividing  through  by  the 
right  member,  we  obtain: 


y 


a2(l  +  sin  a) 
which  may  be  written: 


a2(l  —  sin  a) 
+ 


=  1 


y 


2a2  cos2  ^      2a2  sin^  ^ 


=  1 


(8) 


(9) 


2      ^  2 

where  /3  is  the  complement  of  a.  Equation  (8)  or  (9)  proves 
that  the  locus  is  an  ellipse.  It  is  any  ellipse,  since  by  properly 
choosing  a  and  a  the  denominators  in  (8)  can  be  given  any  desired 
values.  Hence  the  pair  of  parametric  equations  (3)  and  (4),  or 
the  Cartesian  equation  (5)  represents  an  ellipse  with  its  major  axis 
inclined  +  45°  to  the  OX-axis. 

258.  General  Equation  of  the  Second  Degree.  The  general 
equation  of  the  second  degree  in  two  variables  may  be  written  in 
the  standard  form: 

ax""  +  2hxy  +  6?y2  J^  2gx -\- 2Jy  +  c  =  Q  (1) 


438        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§259 

In    the   next    two    sections    we    shall    show   that    the   general 
equation  of  the  second  degree  in  two  variables  represents  a  conic. 
We  shall  be  able  to  distinguish  three  cases  as  follows: 
The  general  equation  of  the  second  degree  represents: 

an  ellipse     iih"^  -  ab  <  0  (2) 

a  parabola    ii  h"^  —  ab  =  0  (3) 

a  hyperbola  ii  h"^  —  ab  >  0  (4) 

To  render  the  above  classification  true  in  all  cases  we  must  classify 

the  "imaginary  ellipse,"  -^  +  v^  =  —  1,  as  an  ellipse,  and  other 
a        0 

degenerate   cases   must   be   similarly   treated.     The   expression 

h^  —  ab  is  called  the  quadratic  invariant  of  the  equation  (1),  so 

called  because  its  value  remains  unchanged  as  the  cui-ve  is  moved 

about  in  the  coordinate  plane.     In  other  words,  as  the  locus  (1) 

is  translated  or  rotated  to  any  new  position  in  the  plane,  and  while 

of  course  the  coefficients  of  x^,  xy,  and  y"^  change  to  new  values,  the 

function  of  these  coefficients,  h"^  —  ab,  does  not  change  value,  but 

remains  invariant.     This  fact  is  not  proved  in  this  book,  but  it  can 

readily  be  proved  by  comparing  the  value  of  h^  —  ab  before  and 

after  the  substitutions : 

X  cos  a  -\-  y  sin  a  —  m  for  x 

y  cos  a  —  X  sin  a  —  n  ior  y 

where  m  and  n  indicate  the  amount  of  the  translation,  and  a  the 

angle  of  rotation. 

259.*  Conies  with  Their  Axes  Parallel  to  the  Coordinate  Axes. ^ 

Let  us  conf  ider  the  equation 

ax^  +  by^  -\-2gx  +  2fy  +  c  =  0  (1) 

If  we  solve  this  equation  for  y  in  terms  of  x,  we  get 

-  /  ±  V-abx^-  2bgx-bc+P  .^. 

y  =  ^ (2) 

1.  We  saw  in  completing  the  squares,  §77,  that  (1)  is  the 
equation  of  an  ellipse  when  a  and  b  are  alike  in  algebraic  signs. 
We  can  now  restate  this  condition  by  saying  that  (2)  is  the  equa- 
tion of  an  ellipse  when  the  coefficient  of  x"^  is  negative.     Note 

^§§259   and  260    are    from  the  correspondence  course  prepared  by  Professor 
H.  T.  Burgess. 


§260]  THE  c6nIC  SECTIONS  439 

that  the  equation  of  a  cu'cle  is  included  as  a  special  case  when 
a  =  b. 

2.  We  saw  in  completing  the  squares,  §87,  that  (1)  is  the 
equation  of  a  hyperbola  when  a  and  b  have  unlike  signs.  We 
can  restate  this  condition  by  sajdng  that  (2)  is  the  equation  of 
a  hyperbola  when  the  coefficient  of  x^  is  positive. 

3.  We  observe  that  (1)  is  the  equation  of  a  parabola 
when  a  =  0.  We  can  restate  this  condition  by  saying  that 
(2)  is  the  equation  of  a  parabola  when  the  coefficient  of  x"^  is 
zero. 

260.  *  The  General  Case.  Write  the  quadratic  in  two  variables 
in  the  standard  form: 

ax''  +  2hxy  +  by''  +  2gx  +  2fy -\- c  =  0  (1) 

I.  We  have  already  seen,  §43,  that  when  and  only  when 
h  =  0  and  a  =  b  the  locus  of  (1)  is  a  circle. 

II.  When  h  is  not  equal  to  zero,  we  have  as  yet  no  knowledge 
of  the  nature  of  the  locus  represented  by  (1),  except  that  it  is 
not  a  circle. 

Let  us  rotate  this  locus  clockwise  through  an  angle  a  and  see  if 
the  equation  can  be  simplified  so  that  the  character  of  the  locus 
represented  by  (1)  can  be  recognized.  Substituting  in  (1)  from 
§256,  we  get 

a(x  cos  a  —  2/  sin  a)  2  +  2h{x  cos  a  —  y  sin  a)  (x  sina-\-  y  cos  a) 
4-  b{x  sin  a  +  2/  cos  a)^  +  2g{x  cos  —  y  sin  a) 

+  2f(x  sin  a  +  y  cos  a)  +  c  =  0  (2) 

If  we  simplify  (2),  we  find  that  the  coefficient  of  the  term  in  xy  is : 

2(6  —  a)  sin  a  cos  a  +  2/i(cos2  a  —  sin^  a)  (3) 

This  term  will  drop  out  of  (2),  if  we  can  find  a  value  for  the  angle 
a  that  will  make  (3)  zero. 
Substituting  in  (3)  from  equations  (1)  and  (2),  §165,  we  get: 

(b  -  a)  sin  2q;  +  2h  cos  2a  =  0  (4) 

From  this  we  find : 

2h 

tan  2a  =    r  (5) 

a  —  b  ^  ' 


440        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§260 

Hence  if  we  choose  a  as  half  of  the  angle  whose  tangent  is  -  _:  7  ^ 
equation  (2)  will  have  no  term  in  xy,  and  it  will  be  of  the  form : 

Ax""  +  Bif  +  2Gx  +  27^?/  +  C  =  0  (6) 

where  A,  B,  G,  F,  and  C  stand  for  long  expressions  in  terms  of 
the  coefficients  of  equation  (1). 

Since  the  loci  of  equations  (1)  and  (6)  are  identically  the  same 
curve,  we  now  see  from  §258  that  the  locus  of  (1)  must  be 
an  ellipse,  hyperbola,  or  parabola. 

III.  We  can  now  devise  a  test  by  which  we  can  tell  immediately 
which  curve  is  represented  by  equation  (1).  If  we  solve  (1)  for 
y  in  terms  of  x,  we  get 


_  -  {hx  +/)  ±\/{h^  -  ab)x''  +  2{hf  -  gb)x  -j-f  -  be 
y  ~  b 

Let  us  now  consider  the  two  equations : 


(7) 


y^  =  -V-b  ^  (^) 


^  Vih''  -ab)x'  +  2{hf  -  gb)x  +  P  -  be  ,q. 

It  is  obvious  that  the  locus  of  (7),  whatever  it  is,  may  be  obtained 
by  shearing  the  locus  of  (9)  in  the  line  (8).  We  must  consider 
the  three  following  cases: 

1.  When  h"^  <  ab  the  coefficient  of  x^  in  (9)  is  negative  and  the 
locus  of  (9)  is  an  ellipse.  Hence  the  locus  of  (7)  is  a  locus  made 
by  shearing  an  ellipse  in  a  line,  and  is  therefore  a  closed  curve. 
The  locus  of  (7)  is  in  this  case  an  ellipse,  for  it  must  be  either  an 
ellipse,  a  hyperbola,  or  a  parabola  by  II,  and  it  cannot  be  either 
a  hyperbola  or  a  parabola  since  it  is  a  closed  curve. 

2.  When  h"^  >  ab  the  coefficient  of  x^  in  (9)  is  positive  and  the 
locus  of  (9)  is  a  hyperbola.  Hence  the  locus  of  (7)  is  a  locus  made 
by  shearing  a  hyperbola  in  a  line,  and  is  therefore  an  open  curve 
with  two  branches.  The  locus  of  (7)  is  in  this  case  a  hyperbola, 
for  it  cannot  be  an  ellipse  or  a  parabola  since  it  has  two  open 
branches. 

3.  When  h^=  ab  the  coefficient  of  x"^  in  (9)  is  zero  and 
the  locus  of  (9)  is  a  parabola.  Hence  the  locus  of  (7)  is  a 
locus  made  by  shearing  a  parabola  in  a  line,  and  is  therefore  an 


j261] 


THE  CONIC  SECTIONS 


441 


open  curve  with  one  branch.  The  locus  of  (7)  is  in  this  case  a 
parabola,  for  it  cannot  be  an  ellipse  or  a  hyperbola  since  it  has 
one  open  branch. 

We  now  state  the  results  in  this  form:     The  locus  of  the  general 
equation  of  the  second  degree  in  two  variables  is  for 
h^  <  ab    an  ellipse 
h^  >  ab    a  hyperbola 
h^  =  ab    a  parabola 

If  we  shear  the  locus  of  (7)  in  any  line  y  =  mx  -\-  b,  the  form 
of  the  equation  is  not  changed.  Hence  the  following  important 
facts: 

The  shear  of  an  ellipse  in  a  line  is  an  ellipse. 

The  shear  of  a  hyperbola  in  a  line  is  a  hyperbola. 

The  shear  of  a  parabola  in  a  line  is  a  parabola. 

If  we  put  (mx)  for  x  and  (ny)  for  y  in  (7),  no  change  will  be 
made  in  the  sign  of  the  coefficient  of  x^;  hence  the  elongation  or 
contraction  {orthographic  projection)  of  an  ellipse,  hyperbola,  or 
parabola  in  any  direction  is  an  ellipse,  hyperbola,  or  parabola. 

261.  Shear  of  the  Circle.  The  effect  of  the  addition  of  the  term 
mx  to  fix),  in  the  equation  y  =  f{x),  has  been  shown  in  §37  to 


Fig.   182. 


-The  Ellipse  Looked  Upon  as  the  Shear  of  a  Circle  OA  in  a  Line 
M'OM. 


be  to  change  the  shape  of  the  locus  by  lamellar  or  shearing  motion 
of  the  xy  plane.  We  usually  speak  of  this  proces's  as ''  the  shear  of 
the  locus  y  =  f(x)  in  the  line  y  =  mx."     When  applied  to  the  circle 


442        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§262 


y  =  +  Va^  —  x^  the  effect  is  to  move  vertically  the  middle  point 
of  each  double  ordinate  of  the  circle  to  a  position  on  the  line 
y  =  mx.     The  result  of  the  shearing  motion  is  shown  in  Fig.  182. 
The  area  bounded  by  the  curve  is  unchanged  by  the  shear. 
The  equation  after  shear  is: 


y  =  mx  ±  Va^  —  x"^  (1) 

This  is  the  same  form  as  equation  (5)  of  §257,  if  we  put 
sin  Of 

m  = and  then  multiply  all  ordinates  by  cos  a.  There- 
cos  a  ^  "^  -^ 

fore  the  curve  of  Fig.  182  is  an  ellipse. 

The  straight  line  ?/  =  mx  passes  through  the  middle  points  of 

the  parallel  vertical  chords  of  the  ellipse 


y  =  mx  -\-  Va^  —  x^  (2) 

The  locus  of  the  middle  points  of  parallel  chords  of  any  curve  is 
called  a  diameter  of  that  curve.  We  have  thus  shown  that  the 
diameter  of  the  ellipse  is  a  straight  line.  Since  the  same  reasoning 
applies  to 


y  =  mx-\-  (6/a)Va2  -  x^  (3) 

which  may  be  regarded  as  any  ellipse  in  any  way  oriented  with 
respect  to  the  origin,  the  proof  shows  that  the  mid-points  of  arbi- 
trarily selected  parallel  chords  of  an  ellipse  is  always  a  straight 
line. 

262.  A  Second  Proof.  The  generality  of  the  preceding  fact 
may  seem  clearer  if  the  ellipse  be  kept  fixed  in  position  while  the 
direction  of  the  set  of  parallel  chords  is  arbitrarily  selected.  Con- 
sider first  the  circle 

a;2  +  2/2  =  a^  (1) 

and  draw  any  set  of  parallel  chords.  Let  the  slope  of  these  chords 
be  s.     Then  the  equation  of  the  chords  is 

y  =^  sx  +  p  (2) 

in  which  p  is  an  arbitrary  parameter,  to  various  values  of  which 
correspond  the  different  chords  of  the  family  of  parallel  chords. 


of  the  ellipse  is: 

x^     y^_^ 

The  parallel  chords 

now 

have  the  equation 

sbx   ,    hp 
y=    a   +  a 

§263]  THE  CONIC  SECTIONS  443 

The  equation  of  the  bisectors  of  all  of  the  chords  is  a  line  through 
the  origin  perpendicular  to  (2),  or: 

X 

y  =  -  s  (3) 

Now  if  the  circle  (1)  and  the  chords  (2)  and  the  diameter  (3)  be 
changed  by  orthographic  projection  upon  a  plane  through  the 
X-axis,  then  the  circle  (1)  becomes  an  ellipse,  while  the  parallel 
chords  and  the  line  through  their  mid-points  remain  straight  lines, 
but  with  modified  slopes.    Let  the  given  orthographic  projection 

multiply  all  ordinates  of  (1),  (2)  and  (3)  by  -  •     Then  the  equation 


(4) 


(5) 

The  equation  of  the  locus  of  the  mid-points  of  the  parallel  chords 
or  the  diameter  is: 

y-'sl  («) 

sh 
Representing  - ,  the  slope  of  (5),  by  m,  equation  (6)  takes  the  form: 

which  is  the  equation  of  the  diameter  of  (4)  that  bisects  the  famQy 
of  parallel  chords  of  slope  m. 

263.  Confocal  Conies.  Fig.  183  shows  a  number  of  ellipses 
and  hyperbolas  possessing  the  same  foci  A  and  B.  This  family 
of  curves  may  be  represented  by  the  single  equation: 

x^  W2 

a^  -  k  "^62-ir^=  1  (1) 

in  which  the  parameter  k  takes  on  any  value  lying  between  0 
and  a^  and  in  which  a  >  b.     li  k  satisfies  the  inequality: 

0  <k  <b^ 
the  curves  are  ellipses.     If  k  satisfies  the  inequality: 

b^  <k  <a^ 


444        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§263 

the  curves  are  hyperbolas.  The  ellipses  of  Fig.  183  may  be 
regarded  as  representing  the  successive  positions  of  the  wave  front 
of  sound  waves  leaving  the  sounding  body  AB;  or  they  may  be 
regarded  as  the  equipotential  lines  around  the  magnet  AB,  of 
which  the  hyperbolas  represent  the  lines  of  magnetic  force. 

Exercises 

1.  Sketch  the  curve: 

y  =  2x  +  \'4:  -  xK 

2.  Draw  the  curve: 

X  =  2  cos  d 

2/  =  2  sin  (0  +7r/6). 


Fig.   183. — Confocal  Ellipses  and  Hyperbolas.     Note  that  the  curves  of 
one  class  cut  those  of  the  other  class  orthogonally. 


3.  Find  the  axes  of  the  ellipse: 

X  =  S  cos  6 

1/  =  3  sin  (d  +7r/4). 

4.  Draw  the  curve: 

y  =  X  ±  \6x  —  x^. 
6.  Draw  the  curve: 

y  =  X  ±  V.T^  —  6a:. 


§263]  THE  CONIC  SECTIONS  445 

6.  Show  that: 

y  =  X  ±  VGx  is  a  parabola. 

7.  Sketch  the  curve: 

y  =  (l/2)x  +  Vl6  -  x^. 

8.  Sketch  the  curve: 

y  =  5x  sin  60°  +  cos  60''V25  -  x^. 

9.  Discuss  the  curve: 

x^/a^  +  y''-/b'^  —  2{xy/ab)  cos  a  =  sin-  a. 

Show  that  the  locus  is  always  tangent  to  the  rectangle  x 
=  +  a,  y  =  ±  b,  and  that  the  points  of  contact  from  a  parallelo- 
gram of  constant  perimeter  4Va^  +  6^  for  all  values  of  a. 

10.  Show  that  x  =  a  cos  (d  —  a),  y  =  b  cos  (d  —  /3)  represents  an 
ellipse  for  all  values  of  a  and  )3. 

11.  Prove  from  equation  (13),  §257,  that  the  distance  from  the 
end  of  the  minor  to  the  end  of  the  major  axis  of  the  resulting  ellipse 
remains  the  same  independently  of  the  magnitude  of  a. 

12.  Show  that  the  following  construction  of  the  hyperbola 
xy^  =  a^  is  correct.  On  the  —  x-axis  lay  off  OC  =  a.  Connect  C 
with  any  point  A  on  the  y-axis.  At  C  construct  a  perpendicular  to 
AC  cutting  the  ?/-axis  in  B.  At  B  erect  a  perpendicular  to  BC  cutting 
the  +  a:-axis  at  D.  Through  A  draw  a  parallel  to  the  x-axis  and 
through  D  draw  a  parallel  to  the  y-axis.  The  two  lines  last  drawn 
meet  at  P,  a  point  on  the  desired  curve. 

13.  Explain  the  following  construction  of  the  cubical  parabola 
a-y  =  x^.  Lay  off  OB  on  the  —  y-axis  equal  to  a.  From  B  draw  a 
line  to  any  point  C  of  the  x-axis.  At  C  erect  a  perpendicular  to  BC 
cutting  the  y-axis  at  D.  At  D  erect  a  perpendicular  to  CD  cutting 
the  X-axis  at  E.  Lay  off  OE  on  the  y-axis.  Then  OE  is  the  ordinate 
of  a  point  of  the  curve  for  which  the  abscissa  is  OC. 

14.  Explain  and  prove  the  following  construction  of  the  semi- 
cubical  parabola,  ay^  =  x\  Lay  off  on  the  —  .x-axis  OA  =  a. 
From  A  draw  a  parallel  to  the  line  y  =  mx,  cutting  the  ?/-axis  in  B. 
Erect  at  B  a  perpendicular  to  AB  cutting  the  ;r-axis  at  C,  and  at  C 
erect  a  perpendicular  to  OC.  The  point  of  intersection  with  y  =  mx 
is  a  point  of  the  curve. 


446        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§263 


Problems  for  Review 

1.  Find  the  approximate  equations  for  the  following  data: 
(a)  Steam  pressure :  v  =  volume,  p  =  pressure. 

(6)  Gas-engine  mixture:  v  =  volume,  p  =  pressure. 


(a) 


(h) 


V 

V 

V 

V 

2 

68.7 

3.54 

141.3 

4 

31.3 

4.13 

115.0 

6 

19.8 

4.73 

95.0 

8 

14.3 

5.35 

81.4 

10 

11.3 

5.94 

71.2 

6.55 

63.5 

7.14 

54.6 

2.  Show  that  p-  =  a^  cos  2d  is  the  polar  equation  of  a  lemniscate. 

3.  When  an  electric  current  is  cut  off,  the  rate  of  decrease  in  the  cur- 
rent is  proportional  to  the  current.  If  the  current  is  36.7  amperes 
when  cut  off  and  decreases  to  1  ampere  in  one-tenth  of  a  second, 
determine  the  relation  between  the  current  C  and  the  time  t. 

4.  Write  four  other  equations  for  the  circle  p  =  2-v/3  sin  0  —  2  cos  6. 

5.  Write  four  other  equations  for  the  sinusoid  ?/  =  sin  a;  —  V  3  cos  x. 

6.  Find  the  angle  that  3a:  +  4?/  =  12  makes  with  Ax  —  3y  =  12. 

7.  From  the  equation 

0  =  6  sin  {2t  -  1°) 

determine  the  amplitude,  period,  and  frequency  of  the  S.H.M. 

8.  A  simple  sinusoidal  wave  has  a  height  of  3  feet,  a  length  of  29 
feet,  and  a  velocity  of  7  feet  per  minute.  Another  wave  with  the 
same  height,  length,  and  velocity  lags  15  feet  behind  it.  Give  the 
equation  of  each. 

9.  Interpret  ri(cos  di  +  i  sin  di)  as  an  operator  upon 
r2(cos  02  -\-  i  sin  62). 

10.  Give  a  rule  for  writing  down  the  value  of  i". 

11.  Calculate: 

(3\/3  -  3i)H  -  1  +  Vdiy       (cos36°  +  isin  36°) (cos 20°  +  i sin 20°) ' 
(2  -f  2V2i)  2(cos  11°  +  i  sin  11°) 

12.  Calc^late:     (1  -  V3t)^i 


§263]  THE  CONIC  SECTIONS  447 

13.  Write  the  inverse  functions  of  the  following: 

(a)  y  =  ax",  (6)  y  =  sin  a:,  (c)  y  =  e*,  {d)  y  =  log«a;. 

14.  Plot  the  amount  of  tin  required  to  make  a  tomato  can  to  hold 
1  quart  as  a  function  of  the  radius  of  its  base.  Determine  approxi- 
mately from  the  graph  the  dimensions  requiring  the  least  tin. 

16.  Find  the  axes  of  the  ellipse  whose  foci  are  (2,  0)  and  (—2,  0), 
and  whose  directrices  are  x  =  ±  5. 

16.  Write  the  polar  equation  for  the  ellipse  in  problem  15. 

17.  Find  the  equation  of  the  hyperbola  whose  foci  are  (5,  0)  and 
(—  5,  0),  and  whose  directrices  are  x  =  ±2. 

18.  Write  the  equation  of  the  hyperbola  in  17  in  polar  coordinates. 

19.  Discuss  the  curve  p(l  +  cos  0)  =  4.  Write  its  equation  in 
rectangular  coordinates. 

20.  Find  the  foci  of  the  hyperbola  2xy  =  a^.     Also  its  eccentricity. 

21.  What  property  of  the  parabola  is  useful  in  designing  automobile 
headlights? 

22.  How  do  you  draw  a  tangent  to  an  ellipse?     To  a  parabola? 

23.  Find  the  equation  of  a  point  whose  distance  from  the  point 
(3,  4)  is  always  twice  its  distance  from  the  line  3a;  +  4?/  =  12.  What 
is  the  locus? 

24.  Give  the  type  of  each  of  the  following  conies: 

(a)  2^2  +  22/2  +  3a;  -  4?/  +  3  =  0. 
(6)  a:2  +  4x2/  +  4^/^  +  a;  -  3?/  +  8  =  0. 
(c)  a;2  +  3x2/  -  32/^  +  3x  -  22/  -  3  =  0. 
{d)  x2  -  5x2/  +  72/2  +  2x  +  32/  +  28  =  0. 

26.  Solve  each  of  the  equations  in  problem  24  for  y  and  explain 
how  the  graphs  may  be  constructed  by  shear. 

26.  A  point  moves  so  that  the  quotient  of  its  distance  from  two 
fixed  points  is  a  constant.  Find  the  equation  of  the  locus  of^^the 
point. 

27.  Evaluate: 

log  10  -  log2  8  +  log7  492. 

28.  Find  the  maximum  and  minimum  value  of  (3  sin  x  —  4  cos  x). 
What  values  of  x  give  these  maximum  and  minimum  values? 

29.  Find  the  equation  of  a  cu-cle  passing  through  the  points  (1,  2) , 
(-1,3)  and  (3,  -2). 

30.  A  sinusoidal  wave  has  a  wave-length  of  tt,  a  period  of  x,  and  an 
amplitude  of  tt.     Write  its  equation. 

31.  Compute  graphically  the  following: 

•- (l+2)(l-t);    (1  +i)  +  (1 -2); 


448        ELEMENTARY  MATHEMATICAL  ANALYSIS      [§263 

f^^]     7  cis  47°  X  6  cis  ( -  14°) ; 

(7  +  6i)";      \/7i  +~3i. 

32.  Prove  by  the  addition  formulas  that: 

sin  (90°  -  t)  =  cos  r 

sin  (90°  +  r)  =  cos  r 

sin  (360°  -  r)  =  -  sin  r  . 

tan  (t  +  270°)  =   -  cotr. 

33.  Sketch  the  curves: 

y  =2- 
y  =  3- 

y    _   20.63a; 

What  property  of  the  exponential  function  do  these  curves  illustrate? 

34.  Sketch  y  =  2''  and  y  =  3^. 

35.  Solve:      x^  +  6x  +  Vx^  +  6a;  +  1  =  1. 

36.  Find  graphically  the  product  of  3  —  2^  by  —  2  +  ^. 

37.  Find  all  the  values  of : 

(cos  0  +  i  sin  6)^;  (cos  d  +  i  sin  0)M;  \/Y;  V IT 

38.  Write  a  short  theme  on  operators,  making  mention  of  (a)  the 
integers;  (6)  (—  1);  (c)  V  —  1;  (d)  cis  6.  Develop  the  rules  for  addi- 
tion, subtraction,  multiplication,  and  division  of  vectors,  and  state 
them  in  systematic  form. 

39.  Show  that 

sin  (a  -{-  b+  c)  =  sin  a  cos  b  cos  c  +  cos  a  sin  b  cos  c 

+  cos  a  cos  6  sin  c  —  sin  a  sin  b  sin  c. 

40.  Sketch  the  curves 

y  =  3- 
1  =  3. 

y     _    gx    +    0-63 

on  the  same  sheet  of  paper.     What  property  of  the  exponential  func- 
tion do  these  curves  illustrate? 

41.  Draw  upon  squared  paper,  using  2  cm,  =  1,  the  curve  y-  =  x. 
By  counting  the  small  squares  of  the  paper  find  the  area  bounded  by 
the  curve  and  the  ordinates  x  =  1/2,  1,  1^,  2,  2 1,  3,  3  ^  4,  .  .  .  By 
plotting  these  points  upon  some  form  of  coordinate  paper,  find  the 
functional  relation  existing  between  the  x  coordinate  and  the  area 
under  the  curve. 


§263] 


THE  CONIC  SECTIONS 


449 


42.  The  latitude  of  two  towns  is  27°  31'.  They  are  7  miles 
apart  measured  on  the  parallel  of  latitude.  Find  their  difference  in 
longitude. 

43.  Solve  S*'^"!  =  2*+i.  Be  very  careful  to  take  account  of  all 
questionable  operations  .     There  are  two  solutions. 

44.  Find  (three  problems)  the  equation  connecting: 


X 

y 

6.8 

19.0 

14.2 

21.6 

21.8 

23.2 

32.0 

26.3 

46.5 

31.5 

65.0 

39.1 

78.0 

47.0 

X 

y 

002 

18.8 

0.04 

19.3 

0.06 

19.6 

0.08 

19.7 

0.10 

20.2 

0.12 

20.5 

0.14 

21.0 

0.16 

21.4 

0.18 

21.8 

0.20 

21.9 

0.22 

22.1 

X 

y 

1.3 

21 

2.0 

25 

2.8 

29 

3.7 

33 

4.3 

35 

5.3 

38 

46.  Find  the  wave  length,  period,  frequency,  amplitude  and  velocity 
y  =  10  sin  {2x  -  30- 
csc^  A 


for 
46.  Prove  that: 


csc2  A  -  2 


sec  2 A. 


47.  Find  the  parametric  equations  of  the  cycloid. 

48.  Find  the  equation  of  the  ellipse,  center  at  the  origin,  axes  coin- 
ciding with  coordinate  axes,  passing  through  the  point  (—  3,  5)  and 
having  eccentricity  3/5. 

49.  Define  the  "logarithm  of  a  number." 

60.  Prove: 

CSC  2x(l  —  cos  2x)  =  sin  x  sec  x. 
CSC  x{\  —  cos  x)  =  ? 

61.  A  S.H.M.  has  amplitude  6,  period  3.  Write  its  equation  if 
time  be  measured  from  the  negative  end  of  the  oscillation.  State  the 
difference  between  a  S.H.M.  and  a  wave. 

62.  Find  by  inspection  one  value  of  x  satisfying  the  following  equa- 
tions : 

29 


450        ELEMENTARY  MATHEMATICAL  ANALYSIS       [§263 

(a)  cos  45°  cos  (90°  -  x)  -  sin  45°  sin  (90°  -  x)  =  cos  x. 

(h)  cos  (45°  -  x)  cos  (45°  +  a:)  -  sin  (45°  -  x)  sin  (45°  +  x)  =  cos  a;. 

53.  Sketch  on  Cartesian  paper: 

2/  =  1* 

?/  =  2*  y  =  log2  X 

2/  =  3*  y  =  logs  X 

2/  =  5*  2/  =  log5  X 

2/  =  10^  2/  =  logio  X. 

54.  Solve  3^  +  2a;  =  1. 

55.  Sketch: 

p  =  a,  p  =  sec  d,  p  =  a  sin  6,  p  =  —  a  sin  ^. 

1 
P  =-,  p  =  a  cos  0,  p  =  —  a  cos  d,  p  =  a  —  a  cos  d. 

p  =  2(  —  cos  d),  p  =  2  cos  0  —  3,       p  =  cos  6  +  sin  0. 

56.  Simplify  the  expression: 

sin   (I  -  T  )   sec   (|  +  r  j  -  sm  (|  +  r  j  sec   (|  -  r  j 

57.  A  point  moves  so  that  the  product  of  its  distance  from  two  fixed 
points  is  a  constant.  Find  the  equation  of  the  locus.  Discuss  the 
curve. 

58.  Simplify  and  represent  graphically : 

(a)  ^^   (6)  (l+i)(l+2i). 

59.  Find  the  velocity  and  frequency  of  the  wave  of  problem  30. 

60.  Find  the  coordinates  of  the  center,  the  eccentricity,  and 
the  lengths  of  the  semi-axes  of:  (a)  x"^  -\-  ^x  -\-  y^  =  7 , 
(6)  x""  +  2x  +.42/2  _  3y  =  0,  (c)  x^  -  X  -  y'  -  y  =  0, 
(d)    a;2  +  a;  +  2/  +  3  =  0. 

61.  Find  the  amplitude,  period,  frequency  and  ^poch  of  the  fol- 
lowing S.H.M.: 

2/  =  7  sin  6^. 

2/  =  6  sin  2Tt. 

2/  =  a  sin  (w<  +  a). 

62.  Find  cis^  d.     Hence  show  that: 

cos  5a;  =  cos^  a;  —  10  cos^  x  sin^  a;  +  5  cos  x  sin*  x. 

63.  Find  graphically  (on  form  Af3)  the  fifth  roots  of  2^  cis  35°. 

64.  Complete  the  following  equations: 

sin  (a  ±  6)  =  ?  tan  2a;  =  ? 

cos  (a  ±  6)  =  ?  cot  2a;  =  ? 


1263]  THE  CONIC  SECTIONS  451 


tan  (a  +  B)  =  ?  sin 


r 


sin  2x  =  t  cos  o^  =  ? 

COS  2x  =  ?  cot  w  =  'i 

65.  Change  the  equations  of  exercises  33  and  40  to  logarithmic 
form.  What  properties  of  logarithms  are  illustrated  by  these 
equations? 

66.  SolvQ  x^  -  x3  +  7x  +  6  =  0. 

67.  Show  that  the  sum  of  the  two  focal  radii  of  the  ellipse  is  constant. 

68.  y  =  —  3^2  +  4i  —  5  and  x  =  5iare  the  parametric  equations  of  a 
curve.     Discuss  the  curve. 

69.  Show  that  [r(cos  6  -\-  i  sin  6)]  [r'(cos  6'  -{■  i  sin  6')]  = 
rr'kos  {6  -{-  d')  +  i  sin  (6  +  6')]. 

70.  Two  S.H.M.  have  amplitude  6  and  period  two  seconds.  The  point 
executing  the  first  motion  is  one-fourth  of  a  second  in  advance  of 
the  point  executing  the  second  motion.  Write  the  equations  of 
motion. 

71.  Show  that: 

sin  5x  =  sin^  x  —  10  sin^  x  cos^o:  +  5  sin  x  cos*  x. 

72.  Prove  that: 

tan  (45°  +  r)  -  tan  (45°  -  r)  =  ^  l^taii^' 

73.  Show  that  the  difference  of  the  two  focal  radii  for  the  hyperbola 
is  constant. 

74.  Find  graphically  the  quotient  of  6  —  2ihy  S  -\-  75i. 

75.  Solve  by  inspection,  for  y: 

sin  (90°  +  hy)  cos  (90°  -  hy)  +  cos  (90°  +  hv)  sin  (90°  -  hv)  =  sin  y. 

76.  Write  the  parametric  equations  for  the  circle,  the  ellipse,  the 
hyperbola. 


CHAPTER  XIV 
A  REVIEW  OF  SECONDARY  SCHOOL  ALGEBRA       . 

300.  Only  the  most  important  topics  are  included  in  this 
review.  From  five  to  ten  recitations  should  be  given  to  this 
work  before  beginning  regular  work  in  Chapter  I. 

With  the  kind  permission  of  Professor  Hart,  a  number  of  the 
exercises  have  been  taken  from  the  Second  Course  in  Algebra, 
by  Wells  and  Hart. 

301.  Special  Products.     The  following  products  are  fundamental: 

(1)  The  product  of  the  sum  and  difference  of  any  two  numbers: 

(x  +  y){x  -  y)  =  x^  -  2/2 

(2)  The  square  of  a  binomial: 

{x  ±  yY  =  x^  ±  2xy  +  2/^ 

If  the  second  term  of  the  binomial  has  the  sign  (  — ),  then  the  mid- 
dle term  of  the  square  has  the  sign  ( — ). 

(3)  The  product  of  two  binomials  having  a  common  term: 

{x  +  a){x  +  &)  ^  x^  -{•  {a  +  b)x  +  ab 
thus  {x  +  b){x  -  11)  =  x2  +  (5  -  ll)a:  +  5(-ll) 

=  x2  -  6x  -  55 

(4)  The  product  of  two  general  binomials: 

{ax  +  b){cx  +  6?)  =  acx^  +  (6c  +  ad)x  +  bd 
thus 

(3a  -  46) (2a  +  76)  =  (3a) (2a)  +  (-  8  +  21)a6  +  (-4  6)  (76) 
=  6a2  +  13a6  -  286^ 

Exercises 

Find  mentally  the  following  products : 

1.  (5a;  -  2yy.  4.  (2m  +  3)(m  +  4). 

2.  (a  +  116)(a  +  36).  5.  {y^  +  42)(7/2  +  4z). 

3.  (a  -  2y)(a  +  12v).  6.  {?>xy  -  7)2. 

452 


SECONDARY  SCHOOL  ALGEBRA  453 


7.  (3w2y  -4)(3w2y  +  4). 

29.  (2  -3s0(5  +2s0. 

8.  (2x  -5)(x  +  4). 

30.  (o26  +  6c)(a26  -  13c), 

9.  (2r2  -7)(3r2  +  5). 

31.  {lp  +  5){lp  -4). 

10.  (p2_3g)(p2_,_7^). 

32.  (a3  +  7)(a3  -  11). 

11.  (a  +  h){a  -  i). 

33.  (3a  +  5)  (7a  -8). 

12.  (^x+52/)(^a:-5i/). 

34.  (1  +8n)(l  -9n). 

13.  {u  -  h){u  -  i). 

36.  (2a  -  6^)  (2a  +  36^). 

14.  {2x+S){hx  +  l). 

36.  (12x  -  i)(9x  -  i). 

16.  (3x2  +  46c)  (3x2  _45c). 

37.  (20  -  162)(3  +2z). 

16.  (2/  -8) (2/  +5). 

38.  (r3  +  16s)  (r3  -  s). 

17.  (X  -  f)(x  -  1). 

39.  (a  -  6x2)  (a  _^  2:2). 

18.  (1  -6s) (3  +2s). 

40.  (4r  +  i^y)(4r  —  5uv). 

19.  (2f  -  7?^2)(3i  _  4^2). 

41.     (6X2    _    1)2, 

20.  (fu-|)(fi.4-i). 

42.  (1  +23n)(5  -  n). 

21.  (3r  -  70  (5r  +20- 

43.    {x^-y^)(x^  +  y^). 

22.  (11x2  _  i)(i2x2  +  1). 

44.  (5a2  -46)(6a2  -  56). 

23.    (22  -  6)(22  +  12). 

46.  {x^y  +  y^x)  (x^y  -  y^x] 

24.    (x+3^2)(^    _2y^). 

46.  (ia  +  10)(2a+l). 

25.  (5m3  -6s2)(5m3  +  s2) 

47.  (9r  +  2s)(3r  -4s). 

26.  (5x  +  |)(5x-i), 

48.  (12x2  +  5)  (4x2  -  3). 

27.  (3x  +  7)(x  -  5). 

49.  (a264  +  4x2)2. 

28.  (4a  -  363)2. 

60.  (a^  -  66)  (a6  +  6«). 

51.  (a +  6)  (a  -  6)(a2  +  62)  C 

a*  +6^). 

52.  (a  -f  6  +  c)2  =  ? 

56.  (a  +  6)4   =  ? 

53.  (a  +  6)3  =  ? 

66.  (a  -  6)4   =  ? 

54    (a  -  6)3  =  ? 

57.  (a  +  6  +  c  +  a)2  =  ? 

68.   5x2  _  ^3^2  +  {2x2  -  (?/2 

+  3x2)  +  5^2}  -x^]  =  ? 

69.  a6  -  [462  _  (2^2  -  62)  - 

{-5a2  +2a6  -362}]  =  ? 

60.  3^/2  -  [22/2  +  (90  -  2?/2)] 

=  ? 

302.  Factoring.  A  rational  and  integral  monomial  is  one  that  is 
made  up  of  the  product  of  two  or  more  arithmetical  or  literal  numbers. 
Thus  10,  7x,  4a6c,  6a^y^  are  rational  and  integral,  but  2a/6,  36v'x 
are  not. 

The  algebraic  sum  of  any  number  of  rational  and  integral  mono- 
mials is  called  a  rational  and  integral  polynomial. 

To  factor  an  algebraic  expression  is  to  find  two  or  more  rational 
and  integral  expressions  which  will  produce  the  given  expression 
when  multiplied  together. 

Next  to  the  removal  of  a  common  monomial  factor  from  all  of  the 
terms  of  a  polynomial,  as,  for  example,  na  -\-  nb  -{-  nc  =  n(a  +  6  +  c), 


454        ELEMENTARY  MATHEMATICAL  ANALYSIS 

the  most  fundamental  cases  of  factoring  are  those  depending  upon 
the  special  products  of  the  preceding  section.     Thus, 

(1)  The  difference  of  two  squares  equals  the  product  of  the  sum  and 
the  difference  of  their  square  roots: 

x^  -  y^  =  (x  -  y)  (x  +  y) 
Thus 

81a8  -  66  =  (9^4  _  63)  (9^4  +  53) 

(2)  A  trinomial  is  a  perfect  square  when,  and  only  when,  two  of  its 
terms  are  perfect  squares  and  the  remaining  term  is  twice  the  product  of 
their  square  roots. 

To  find  the  square  root  of  a  trinomial  perfect  square,  take  the  square 
roots  of  each  of  its  two  perfect  square  terms  and  connect  them  by  the 
sign  of  the  remaining  term. 

Thus,  9a2  —  24a6  +  I662  is  a  perfect  square,  since  VQo^  =  3^, 
V1662  =  46  and  24a6  =  2  (3a)  (46). 

Also  9a2  +  30a  +  16  is  not  a  perfect  square,  for  30a  does  not  equal 
2(3a)(4). 

(3)  Trinomials  of  the  form  x"^  -\-  px  -\-  q  can  he  factored  when  two 
numbers  can  he  found  whose  product  is  q  and  whose  sum  is  p. 

Thus  x2  -  4a:  -  77  =  (x  +  7) (a:  -  11),  for  7(-  11)  =  -  77  and 
(+7)  +  (-11)  =  -4. 

(4)  Trinomials  of  the  form  ax^  -{-  hx  -\-  c,  if  factorable,  may  he  fac- 
tored in  accordance  with  the  properties  of  the  special  product  (4),  §301. 
In  the  product 

ax  +  6 
ex  -{-  d 
acx^  -\-  (be  -\-  ad)x  +  hd 

the  terms  acx"^  and  bd  are  called  end  products  and  hex  and  adx  are 
called  cross  products.  This  most  important  case  of  factoring  is  best 
learned  from  the  consideration  of  actual  examples. 

Factor  21^2  +  5a:  -  4. 

From  the  term  2\x^,  consider  as  possible  first  terms  7a:  and  3a;, 
thus  (7a:  )(3x  ).  For  factors  of  (—  4),  try  2  and  2,  with  unlike 
signs,  and  signs  so  arranged  that  the  cross  product  with  larger  absolute 
value  shall  be  positive;  thu^  (7a:  —  2) (3a:  +  2).  This  gives  middle 
term  8a:;  incorrect.  For  ( —  4)  try  4  and  1,  with  signs  selected  as  be- 
fore; thus,  (7a:  -  l)(3x  +  4).  Middle  term  25x;  incorrect.  Try 
i^x  +  4)(3x  —  1).     Middle  term  5a;;  correct. 


SECONDARY  SCHOOL  ALGEBRA 


455 


(5)  The  difference  of  two  cubes:  x 
Thus 


(x  -  y)ix^  +  xy  +  2/2). 


27x3  _  2/6  =  (3x)3  _  (2/2)3 

=  (3x  -2/2)  (9x2  +  3x2/2  _|_2/4) 

(6)  The    sum    of    two    cubes:  x^  +  y^  =  {x  +  y){x^ 

Thus  125a3  +69  =  (5a) '  +  (63)3 

=  (5a  +  63)(25a2  -  5a63  +  h^) 


xy  +  1/2). 


303.  To  factor  a  polynomial  completely,  first  remove  any  monomial 
factor  present;  then  factor  the  resulting  expression  by  any  of  the  type 
forms  which  apply,  until  prime  factors  have  been  obtained  throughout. 
Thus, 


(a)  5a6  -  566  =  5(a6  _  6«)  =  5(a3  -  h^){a^  +  6^) 
=  5(a  -  6)(a2  +  a6  +  62) (a  +  6)(a2 
(6)  42ax2  +  lOax  -  8a  =  2a(21x2  +  5x  -  4) 
=  2a(7a;  +  4)(3x  -  1) 


a6  +  62) 


Exercises 


Factor  the  following  expressions: 


■^^u^ 


9^8   _  4^/6. 

25x*  -  1. 
81  -  4x2. 
1  -  64a264c«. 
x^  —  y^. 
225    -  a\ 
121x2    -  1442/2. 

49m4  -  36x2?/222. 
169  -7%a4x2. 
4x2  -  20x  +  25. 
9a2  +  6a6  +  62. 
a262  -  17a6c  -  60c2. 


1. 

2. 
3. 

4. 

6. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 

14.  r*  -  llr2  +  30. 

15.  1662  +  306  +  9. 

16.  81^2  +  ISOuw  +  100y2. 

17.  36a2  -  132a  +  121. 

18.  x'^y^  -  4xi/2  +  4. 

19.  a262  -  2a6  -  35. 

20.  w6  +  1*3  -  110. 

21.  0^62  -  14a26  +  49. 


\.  x2  +  6x  -  27. 


23.  c3  -64^3. 

24.  8x3-1. 

25.  1  -  13^  -  68^2, 

26.  x4  -  6x26  -  5562. 

27.  0^2  —  4awy  —  45av2. 

28.  28a2  -  a  -  2. 

29.  3s2  -  17s«  +  24«2. 

30.  15r2  -  r  -  6. 

31.  42/2  _  32/  _  7. 

32.  64w8  -  27x3. 

33.  6ar  —  3as  +  4aL 

34.  a2  +  2a  -  35. 

36.  9x2  +  i2xy  -  322/2. 

36.  a2  +  10a6  +  2562. 

37.  625x22/2  -  ^. 

38.  3cd2/2  _  ^cdy  -  30cd. 

39.  4ax2  -  25a2/4. 

40.  32/3  +24. 

41.  4x2  _  27x  +  45. 

42.  6x2  +  7x  _  3. 


456        ELEMENTARY  MATHEMATICAL  ANALYSIS 

43.  ^V  -  1.  68.  2am2  -  50a. 

44.  lOx^y  -  5x22/2  -  5xy^.  59.  72  +  7x  -  49a;2. 

45.  m2n2  +  7mn  -  30.  60.  31rc2  +  2Sxy  -  Sy\ 

46.  rc2  -  Sxy  -  70y\  61.  24a2  +  26a  -  5. 

47.  mx2  +  7ma;  -  44m.  62.  1  -  Sxy  -  10Sx^y\ 

48.  a;3  -  3x2  _  iqSx.  63.  x2  -  14mx  +  40w2. 

49.  x8  -  ?/8.  64.  26  +  10a&  -  28a2b. 

50.  x^  -  bx^y  -  24?/2.  65.  c^  +  27^3. 

51.  8n2  +  ISn  -  5.  66.  ^x^y  -  27xyK 

52.  3x4  _  12.  67.  TVa^'2/2  -  ^^x^?/*. 

53.  9m2  -  4:2mt  +  49^^  68.  ^9n^y  -  196n2?/3. 

54.  10x2  -  39x  +  14.  69.  x2  -  16x  +  48. 

55.  12x2  +  llx  +  2.  70.  x2  +  23x  -  50. 

56.  36x2  +  12a;  _  35.  71.  o%4  +  ^la^^^  4.  30. 

57.  x3  -  Sy\  72.  9x2  _|_  ^'j^y  +  42/2. 

304.  General  Distributive  Law  in  Multiplication.  From  the  mean- 
ing of  a  product,  we  may  write 

(a  +  64-c+.    .    .){x  +  y  +  z  +.    .    .)  =  ax  +  bx  +  ex  + .  .  . 

+  ay  +  hy  +  cy  +.  .  . 

+  az  +  hz  -\-  cz  + .  .  . , 
etc. 

Stating  this  in  words :  The  produet  of  one  polynomial  by  another  is  the 
sum  of  all  the  terms  found  by  multiplying  each  term  of  one  polynomial 
by  each  term  of  the  other  polynomial. 

To  multiply  several  polynomials  together,  we  continue  the  above 
process.  In  words  we  may  state  the  generalized  distributive  law  of 
the  product  of  any  number  of  polynomials  as  follows : 

The  product  of  k  polynomials  is  the  aggregate  of  all  of  the  possible 
partial  products  which  can  be  made  by  multiplying  together  k  terms,  of 
which  one  and  only  one  must  be  taken  from  each  polynomial. 
Thus, 

{a  +  b+c  +  .    .    .){x-\-y  +  z+.    .    .){u  +  v  +  w+.    .    .) 
=  axu  +  axv  +  •    •    •  +  o,yu  +  ayv  4-  •    .    .  +  clzu  +  azv  +    .    .    . 
+  bxu  +  bxv  +  .    .    .   +  byu  +  byv  +  .    .    .  +  bzu  +  bzv  +  .    .    . 
+  cxu  +  cxv  +  .    .    . 
+  .    .    .,  etc. 

//  the  number  of  terms  in  the  different  polynomials  be  n,  r,  s,  t.    .    . 
respectively,  the  total  number  of  terms  in  the  product  will  be  nrst   .    .    . 
The  student  may  prove  this. 


SECONDARY  SCHOOL  ALGEBRA  457 

306.  The    Fundamental  Theorem  in  the   Factoring    of  x**  ±  a**. 

The  expression    (x^  —  a")  is  always  divisible  by  {x  —  a). 
Write  X"  —  a"  =  oj"  —  ax^~^  +  ax"*'^  —  a" 

=  x'^'^x  —  a)  +  aix""'^  —  a''~i) 

Now  if  (x**"!  —  o**"^)  is  divisible  by  (x  —  a),  then  plainly 
.T"~Hx  —  a)  +  a{x^~^  —  a^~^)  is  also  divisible  by  {x  —  a).  But 
this  last  expression  equals  (x"  —  a"),  as  we  have  shown.  Therefore, 
if  (x  —  a)  exactly  divides  {x"'~^  —  a"^'^),  it  will  also  exactly  divide 
(x**  —  a"). 

But  {x  —  a)  will  exactly  divide  (x^  —  a^),  therefore  it  will  divide 
(x*  —  a*),  and  since  (x  —  a)  exactly  divides  {x*  —  a^)  it  will  exactly 
divide  {x^  —  a^),  and  so  on. 

Therefore,  whatever  positive  whole  number  be  represented  by 
n,  {x  —a)  will  exactly  divide  (x**  —  a^). 

We  see  that  (x  —  a)  is  one  factor  of  (x**  —  a").  The  other  factor 
of  (x**  —  a**)  is  found  by  actually  dividing  (x**  —  a**)  by  (x  —  a). 
Thus 

(x'*  —  a")  =  (x  —  a)(x'*-i  +  ax''-^  +  a^x''"^  +..._}_  a'^-^x  +  a""^) 

The  student  may  show  that  (x  +  a)  divides  x«  +  a"  if  n  be  odd,  and 
divides  x"  —  a"  if  n  be  even. 

3C6.  Quadratic  equations  are  usually  solved  (a)  by  factoring,  (6) 
by  completing  the  square,  or  (c)  by  use  of  a  formula. 

(a)  To  solve  by  factoring,  transpose  all  terms  to  the  left  member  of 
the  equation  and  completely  factor.  The  solution  of  the  equation  is 
then  deduced  from  the  fact  that  if  the  value  of  a  product  is  zero,  then 
one  of  the  factors  must  equal  zero.     Thus 

(1)  Solve  the  equation 

x2  +  54  =  15x 

Transposing  x^  —  15x  +  54  =0 

Factoring  (x  -  9)(x  -  6)  =0 

x-9  =  0ifx  =  9 

X  -  6  =  Oif  X  =  6 

Hence  the  roots  of  the  equation  are  9  and  6. 

Check:     Does  (9)^  +  54  =  15  X  9? 
Does  (6)2  +  54  =  15  X  6? 

(6)  To  solve  by  completing  the  square,  use  the  properties  of 
(x  ±  a)2  =  x^  ±  2ax  +  a^,  as  follows: 


458        ELEMENTARY  MATHEMATICAL  ANALYSIS 

(2)  Solve  x2  -  12a:  =  13. 

Add  the  square  of  1/2  of  12  to  each  side 

x2  -  12x  +  36  =  49 

Take  the  square  root  of  each  member 

re  -  6  =  +  7 

Hence 

a;  =  6  +  7  =  13 
a;  =  6  -  7  =  -1 

Check:     Does  (13)2  -  12  X  13  =  13? 

Does  (-1)2  -  12  X  (-1)  =  13? 

Since  in  general  {x  —  a){x  —  h)  =  x"^  —  (a  +  h)x  +  ah,  we  can  check 
thus: 

Does  13  +  (-  1)  =  -  (-  12)? 

Doesl3(-  1)  =  -  13? 

(3)  Solve  x2  -  20x  +  97  =  0. 

Transpose  97  and  add  the  square  of  1/2  of  20  to  each  side : 
a;2  -  20a:  +  100  =  -  97  +  100  =  3 
Take  the  square  root  of  each  number: 

x  -  10  =  ±  \/s 
Hence 

xi  =  10  +  a/s" 

a:2  =  10  -  Vs 

Check:     Does  a:i  +  0:2  =  —  (—  20)? 
Does  xiX2  =  97? 

(c)  To  solve  by  use  of  a  formula,  first  solve 

ax2  +  6x  +  c  =  0  (1) 


The  roots  are 

-  b  ±  Vb^  -  ^ac 

X  = ^ 

2a 


(2) 


For  a  particular  example,  substitute  the  appropriate  values  of  a,  b, 
and  c.     Thus: 

(4)  Solve  2a:2  -  3a:  -  5  =  0. 

Comparing  the  equation  term  by  term  with  (1)  we  have 

a  =  2,h  =  -3,  c=  -5 


SECONDARY  SCHOOL  ALGEBRA  459 

Substitute  these  values  in  the  formula  (2) 


Therefore 


.  _  -(-3)±  V(-3)^-4(2)(-"5) 
2(2) 
3  ±  7 


xi  =  5/2,  X2  = 


Check:     Does  xi  +  xa  =  —  h/a  =  3/2? 
Does  XiXi  =  c/a  =  —  5/2? 

Exercises 

Solve  the  following  quadratics  in  any  manner: 

1.  x2  +  5x  +  6  =  0. 

2.  x^  +  4x  =  96. 

3.  x^  =  110  +  X. 

4.  x^  +  5x  =  0. 
6.  6x^  +  7x  +  2  =  0. 

6.  8x2  -  lOx  +  3  =  0. 

7.  a:2  +  mx  -  2m^  =  0. 

8.  3f2  -  <  -  4  =  0. 

9.  10r2  +  7r  =  12. 

10.  a;2  +  2ax  =  h. 

11.  a;2  +  4x  =  5. 

12.  x2  +  6a:  =  16. 

13.  2x^  -  20x  =  48. 

14.  x^  +Sx  =18. 
16.  x^  +  5x  =  36. 

16.  3x2  +  6x  =  9. 

17.  4x2  _  4a;  =  8. 

18.  x2  -  7x  =  -  6. 

19.  x2  -  ox  =  6a2. 

20.  x2  -  2ax  =  3a2 

21.  x2  -  X  =  2. 

22.  x2  +  X  =  a2  +  a. 

23.  x2  -  lOx  =  -  9. 

24.  2x2  -  15x  =  50. 

25.  x2  +  8x  =   -15. 

26.  3x2  _|_  12a:  =  36. 

27.  2x2  +  lOx  =  100. 

28.  x2  -  5x  =  -  4. 


29. 

3x2 

-  12ax  =  63a2. 

30. 

4x2 

-  12ax  =  16a2. 

31. 

X2 

-X  =  6. 

32. 

X2 

+  7x  =  -  12. 

33. 

'x2 

-  5x  =  14. 

34. 

X2 

+  X  =  12. 

35. 

X2 

-  X  =  12. 

36. 

X2 

=  6x  -  5. 

37. 

X2 

=  -  4x  +  21. 

38. 

X2 

=  -  4x  +  5. 

39. 

X2  . 

-1-  5x  +  6  =  0. 

40. 

X2 

+  llx  =  -  30. 

41. 

X2 

-  7x  +  12  =  0. 

42. 

X2  • 

-  13x  =  30. 

43. 

3x2 

+  4x  =  7 

44. 

3x2  +  6x  =  24. 

45. 

4x2 

-  5x  =  26. 

46. 

5x2 

-  7x  =  24. 

47. 

2x2 

-  35  =  3x. 

48. 

3x2 

-  50  =  5x. 

49. 

3x2 

-  24  =6x. 

50. 

2x2 

-  3x  =  104. 

51. 

2x2 

+  lOx  =  300. 

52. 

3x2 

-  lOx  =  200. 

53. 

4x2 

-  7x  +  i  =  0. 

54. 

h' 

-h=-  il 

55. 

9x2 

+  6x  -  43  =  0. 

56. 

18x2  -  3x  -  66  =  0 

460        ELEMENTARY  MATHEMATICAL  ANALYSIS 

57.  ix2  -  Sx  +  \l  =  0.  59.  2x2  _  22a;  =  -  60. 

KO    ^'        3^    I   o       r.  ^^-  3^'  +  7x  -  370  =  0. 

58-   4   -  "2   +  2  =  0-  61.  5a:2  _  |_^  _  _7^  =  q. 

««    ^^       ^     ,    1 
^2-3-2+6=0. 

63.  x2  +  2x  +  1  =  6x  +  6.  69.  s^  =  5s  +  6. 

64.  x2  -  49  =  10(rc  -  7).  70.  r2  +  3r  =  4. 

65.  2x2  +  60x  =  -  400.  71.  2s2  +  4as  -  c  =  0. 

66.  a2  +  7a  +  7  =  0.  72.  x^  +  6ax  -  5  =  0. 

67.  z^  =  dz  +  2.  73.  x2  -  lOax  =  -  9a\ 

68.  r  =  r2  -  3.  74.  cx2  +  2dx  +  e  =  0. 

75.  2x2  +  6x  -  n  =  0. 

76.  2/2  +  2^  =  3  83^  4^2  -  3x  =  3. 

77.  x2  =  5  +  |x.  84.  9^2  +  4^  =  6. 

78.  ?^2  _  6^  _  1  =  0.  85.  5(x2  -  25)  =  X  -  5- 

79.  ^2  +  1^  =  |.  86.  9^2  +  I8w  +  8  =  0 

80.  r2  -  I  =  f  r.  87.  x2  +  px  +  q. 

81.  s2  -  |s  =  -2/-.  88.  x^  -  8x2  +  15  =  0. 

82.  3r2  -  2r  =  40 89.  u^  -  29u^  +  100  =  0. 

^"^"l-si  93.^+^  =  3. 

91.  2j,  +  |=#-  94"-"       "-^"--3 


4i/ 


5- 

n  — 

X 

■3 

■^8-x 
n+4 

n  - 
8x 

■2 

7x  +  ll 

96.  2*  24      .    .        „ 


X        X  -  2  ^ 
97.  x«  -  35x3  +  216  =  0. 


.s..^m__,,.        3a.(.^g^--(.^g+... 


100.  M  +  ^  =  a  +  - 
2^  'a 


307.  The  Definitions  of  Exponents. 

(1)  n  a  positive  integer:  a**  =  aaa   .    .    .  to  n  factors. 

(2)  n    and    r    positive    integers:    a^"  =  \/a  and  a"/'  =  (*>/a )" 

=  v^. 

(3)  a"  =  1. 

(4)  n  any  number,  positive  or  negative,  integral  or  fractional: 
a""  =  1/a", 

308.  The  Laws  of  Exponents.     For  n  and  r  any  numbers,  positive 
or  negative,  integral  or  fractional : 

(1)  a'^a''    =  a"'^'",  or  law  for  multiplication  and  division. 


SECONDARY  SCHOOL  ALGEBRA  461 

(2)  (a")'  =  a"*",  or  law  for  involution. 

(3)  a"^'*   =  (ab)**,  or  distributive  law  of  exponents. 

Note:  The  student  must  distinguish  between  —  a"  and  (—  a)". 
Thus  -  8^^  =  -  2,  and  (-  8)^^  =  -  2,  but  (-  3)^  =  9  and  -  3^  = 
-9. 

Exercises  1 

Use  the  definitions  of  exponents  (1),  (2),  (3),  (4)  §307,  and  the  laws 
of  exponents  (1),  (2),  (3),  §308,  and  find  the  results  of  the  indicated 
operations  in  the  following  exercises. 

1.  xi2xi3. 

2.  a^'^a^''. 

3.  x^^+ia;". 

4.  b^b^+s. 

5.  m"+^m'*~^. 

6.  a"-2a3+'*. 

7.  x''-'-+^x\ 

8.  m5«m-2«. 


Write  each  of  the  following  sixteen  expressions,  using  fractional 
exponents  in  place  of  radical  signs : 

1.  Va.                  5.  \/^.  9.  V^^  13.  v'^';^^ 

2.  VaK                6.  (V^)6.  10.  {\/x)'.  14.  (v^^-^)'- 

3.  Va^.                7.  >7a^  11.  v^x*.  16.  Va^  -  b^. 

4.  >7^                 8.  (v/a)^  12.  (^-)t.  16.  ^(a+6)'. 

Find    the    numerical    value  of    each    of    the  following    sixteen 
expressions : 


9. 

a:i3  ^  xK 

17.  (a7)3. 

10. 

X8    -T-  X^K 

18.  (a*) 6. 

11. 

Q^Zn    ^   (jn 

19.  (-a62)3. 

12. 

gn+6    jL.  qZ^ 

20.  (a2^2)5. 

13. 

102r+3    ^   10^ 

21.  (6-)  2. 

14. 

^r+6    ^  ^r+3_ 

22.  (-a^feO^ 

15. 

W"+'"    -T-   W'*"'". 

23.  {a^h^y. 

16. 

/       9\ 

24.  (r'»s'*)P. 

-     ©"• 

27. 

Exercises  2 

17.  4^. 

21.  625^. 

26.  81^. 

29.  256^ 

18.  27^. 

22.  64«. 

26.  125^ 

30.  64^. 

19.  9*. 

23.  216i 

27.  32*. 

31.  512^. 

20.  16^ 

24.  16^. 

28.  81^. 

32.  128^. 

Write  each  of  the  following  expressions  m  two  ways,  using  radical 
signs  instead  of  fractional  exponents : 


462        ELEMENTARY  MATHEMATICAL  ANALYSIS 


33.  J. 

37.  J.                  41.  r^. 

45.  al. 

34.  it. 

38.  h^.                  42.  x"^. 

46.  5^. 

36.  rJ. 

39.  e«.                   43.  y\ 

n+l 

47.  X  ■•  . 

36.  x^. 

40.  /i^.                   44.  a~«. 
Exercises  3 

r+i 

48.  a  '  • 

Perform  the  indicated  operations  in  each  of  the  following  exampl< 
by  means  of  the  laws  of  exponents. 

1.  J  X  a^. 

=  ««. 

2.  xi  X  xK 

4.  x^  X  a:^. 

6.  xTn  X  ai;» 

3.  x^  X  xt'^ 

5.  a^  X  a*. 
8.  J  -r  a*. 

a^  -^  at  =  a'^t  =  a^-^x 

2                  r 

7.  a  n  X  a2^ . 

=  a^V    ^ 

9.  h^  -T-  h^. 
10.  m^  -^  m^^. 

12.  9J  -f-  a*. 

15.    (a^)«. 
(a^ )  8  =  all  =  a^^. 

13.  6a*  -^  Sa^. 

14.  a6T  -  a'  hTr. 

16.  (a^)T^.                       18.  (a"'")^.  20.  [(a:")v]7-. 

17.  (/it)!;                         19.  (oA)4.  21.  (a;iiSi)r. 

22.  {Jx^y^)K 

(a%V)*  =  (at)*(a;^)*(2/^)5  =  aM^o. 

23.  (a^b^)^.                     25.  (36a4a:2^3)i,  27.  (32x^/)i 

24.  (adi)t.                      26.  (a^x^yiy^  28.  (^a663c)«. 

aAi 


29. 

\b^ 


(af\?  ^  (at )6  _  ah_ 
Vy     (6?)?  ~  6^ 

36.  (a^  +a*  +  l)(a*  +a  -  a^). 


SECONDARY  SCHOOL  ALGEBRA  463 


arrange  the  work  thus: 

J  +  J  +  1 

a^  +  J-  +  J 

a^  +  ai  +  a 
—  a^  —  a  - 

-a* 

a3  4-  2a3  +  a^         -  a^ 

37.  {x  +  2yi  +  Sy^){x  -  2yi  +  3y^). 

38.  {x^  +yhix^  -yh- 

39.  {J  -  Sah^  +  4:ah  -  ah^iJ  -  2ah^). 

3  1^  i  i  i 

40.  (a^  -  2a«+ 3a'*)(2a«  -  a«). 

Exercises  4 

Find  the  numerical  value  of  each  of  the  following : 

4.  10-6.  7.  2-4. 

5.  1-1  8.  16-^ 

6.  2-2.  9.  Sl-^ 
5  5-2 

15.   /     ^^-3-  17. 


1. 

2-1. 

2. 

4-2. 

3. 

(-2) 

13. 

1 
2-1' 

10. 

1024-^. 

11. 

512-3. 

12. 

625-1 

19. 

16-1 

5-2 

20. 

7-1 
49-i 

14.  ^.  16.  ^,  18.  1^ 

Write  each  of  the  following  expressions  without  using  negative 
exponents : 

21.  x-2.  25.  5a-5.  29.  {x  +  y)-\  33.  2a3a;-2i/-i. 

22.  X27/-2.  26.  3a-26-2.  30.  (- a;)-3.  34.  (- a2)-3, 

««      1  „«     2a-2  «.  a;^  „.      a-ffee 

23.  --,•  27.  TTTir-,-  31.  :—.•  35. 


x-^  -••  362a;-3  "-  2/-5  -"-•  3a,-i6-i 

«.    »^"'  ««     «'&"*  ««    3af6-i  „^  3a2&-2c-4 

24.  -^-  28.       -  _■■•  32.  5 —  36.  -  _,,_,  _,• 

Write  each  of  the  following  expressions  in  one  line: 

«»     1  ««     ab^  .-     3a;%-»  ^^  -r-^^a 

37.  -^-  39.  -rrr  41.   ^  _,,^  •  43. 


38.  — •  40.  o— o,--,'  42.     ...__,  44.    — tttt; 


a"  3a-2y-3 


aj-ry 


*^*  (a-6)-fc-|  ^®-  "^^  +  x^  +  ;."+  ^^' 


464        ELEMENTARY  MATHEMATICAL  ANALYSIS 

Exercises  6 

Perform  the  indicated  operations  in  each  of  the  following  by  means 
of  the  laws  of  exponents. 

1.  a*  X  a-K  4.  Sa-*  X  Sa^.  7.  m"^  X  m,-^. 

2.  ri2  X  r-io.  5.  w-i  X  u^.  8.  Qax-^  X  \hx\ 

3.  c-8  ^  c-5.  6.  a;5  -^  a;"".  9.  a-^b""  ^  ah-\ 

10.  (-  7a-36-2)(-4a26-J)(a-262a;-i). 

11.  {2ah-i){a-h'^ -  \ahi  +  a^-'). 


12.  la-^h-^c- 

3H-8a 

-26-3C-4. 

23.  (a-''62c6)-3. 

13.  56a;62/-724 

-^  7a:- 

-ii/-%-4. 

24.  (x^?/*)-i2. 

14.  ISa-i&tc- 

-5  -^  6a263c-5. 

25.  (8x3^-6)-3. 

15.  6a;32/-*26 

^  2a:- 

V,-i. 

26.   (x-U*)'^". 

16.  (a-3)2. 

17.  (a-2)-5. 

27.  (a-56-io)-*. 

18  (a5)-2. 

28.  (-  la:3)-4. 

19.  (n^)-3. 

29.  (-  a-^)-\ 

30.  (-a«)-*. 

31.  (-8-3b^c-^)-^. 

20.  {r-^r^, 

21.  (c-8)^. 

22.   {abc)-\ 

32.  (-la:3t/-»)3  . 

33.    0- 

"•(f 

:)■'      »■  f^')' 

-•    p.-' 

"■(1 

m-    »■  (my 

-  (P):^ 

-s 

■'b\-'               41     /    "''>-''    \ 

y-)-         "-(x-v-ij 

42.  (a2a:-i+3a3a: 

-2)  (4a-i  -  5x-i  +  6aa:-2) 

4a-i 

-  5a:-i  +  6aa:-2 

a2a;-i 

+  3a3a:-2 

4aa:~ 

1  -  5a2a:-2 

+  QaH-^ 

12a2x-2 

-  15aH-^  +  lSa*x-* 

4ax-i  +  7a2a:-2  -    9a3a:-3  +  18a^a:-* 

43.  (2x-"  -dx  +  4a:^)(a:-^  -  2x-^  +  3a:-^). 

44.  (x-^  -  2x-h^  +  2/*)(a:-^  -  yh- 

45.  (3x*  -fx*  +  4)  X2a:-i 

46.  (x-*  +  x-i  +  l)(a:-3  -  1). 

47^-^  +  y-^)ix-'^  -  y-')' 
48.  {x^y'+yhixi  -  y-^- 


SECONDARY  SCHOOL  ALGEBRA       465 

49.  {2J-  Saxhi^a-^  +  2x-^){aJx^  +  90"^). 
60.  {x-^  -  x-^y^  +  x-^y  -  y^)   ^  {x'^  -  y^). 

x-^  —  y^)x-^  -  x-^y^  +  x-^y  —  y^{x-^  +  y 
x~^  —  x~^y^ 

x-^y  -  y^ 


51.  (x-3  +  2x-2  -  3x-0  ^  (x-2  +  3x-i). 

309.  Reduction  of  Surds  or  Radicals. 

1.  If  any  factor  of  the  number  under  the  radical  sign  is  an  exact 
power  of  the  indicated  root,  the  root  of  that  factor  may  be  extracted  and 
written  as  the  coefficient  of  the  surd,  while  the  other  factors  are  left 
under  the  radical  sign. 

(1)  Thus,  Vs  =  V'fy^'2 . 

=  \/W~2 
_  =  2\/2 ■ 

(2)  Also,  -^81  =  -^27  X  3 

=  '^27</"3 
___  =3^3^ 

(3)  Also,  ^IQax^  =  VSx^  X2ax 

=  i/3x^J/2ax 
=  2xV2ax 

2.  The  expression  under  the  radical  sign  of  any  surd  can  always  be 
made  integral. 


;/h;|x«  -'- 


3         9  \27 


^ 


-  X  18 
27 


=  Wis 

3 


(2)  Also  ^!=^Ix?=^ixl4 


i</u 


30 


466        ELEMENTARY  MATHEMATICAL  ANALYSIS 

3.  We  may  change  the  index  of  some  surds  in  the  following  manner: 

(1)  Thus,  </4  =  VVl 

_  =  V2 

(2)  Also,  V^IOOO  =  WlOOO 

=  a/Io 

(3)  Also,  \/256c2a8  =  \/V256^hi^ 

=  \/lQca^ 

A  surd  is  in  its  simplest  form  when  (1)  no  factor  of  the  expression 
under  the  radical  sign  is  a  perfect  power  of  the  required  root,  (2)  the 
expression  under  the  radical  sign  is  integral,  (3)  the  index  of  the  surd  is 
the  lowest  possible. 

Methods  of  making  the  different  reductions  required  by  this  defini- 
tion have  already  been  explained.     We  give  a  few  examples. 


(1)  Simplify  ^1^. 

\sV''- 

=  W' 

(2)  Simplify  Ji''?. 

= 

=  ±V.a, 

=Vf 

= 

-Iv^o 

= 

-l^T. 

(3)    Simplify  |^?| 

5«/512_ 
2X125' 

5  /i 
"2V5 

■4 

=  Vio 

SECONDARY  SCHOOL  ALGEBRA  467 

In  any  piece  of  work  it  is  usually  expected  that  all  the  surds  will 
finally  be  left  in  their  simplest  form. 

Exercises 
Reduce  each  of  the  following  surds  to  its  simplest  form : 

\5_  \£  \81  \64x6 

'^ii     ';/i    '-^i    '-nR.- 

9.  Simplify  v'12  +  A  V75  +  6V^- 

10.  Simplify  1  +  Vs  +  a/2  -  V27  -  Vl2  +  \/75. 

11.  Simplify    ^^21  +  7\/2    X  ^^21  -  7V2. 

12.  Find  the  value  of  x^  -  6j;  +  7  if  x  =  3  -  a/3. 

13.  Find  the  value  when  x  =  a/3  of  the  expression 

2a;-  1    _    2a;  +1 
(a;-  1)2  ~  (a; +  1)2' 

14.  Find  the  value  of 

(35VrO  +  77a/2  +  63A/3)(A/iO  +  V2  +  a/3). 
Solve  and  check  each  of  the  following  equations: 

15.  a/^T5_=  4. 

16.  V2x  +  6=4. 

17.  \/lOx  +  16  =  5. 

18.  a/2x  +  7  =  V5x  -  2. 

19.  14  +  3/4a;  -  40  =_10. 

20.  a/163;  +9  =  4A/4a;  -  3. 

21.  VY  +  o;  =  I  +  a/^- 

22.  Vx^^A/a;  -  5  =  a/5. 

23.  a/^::iJ  =  A/a;-  14  +  1. 

24.  Va;  -  7  =  Vx±  1-2. 

25.  a;  =  7  -  \/x^  -  7. 

26.  \/a;+20  -  Vx  -  1  -3=0. 

27.  Vx  +  3_+  VSx-_2  =  7.  _ 

28.  a/2x  +  1  +  a/x  -  3  =  2Vx. 
20a; 


Logarithms 


lo     |i    |2l3|4     Is     16|7l8l9|i23U     S     6l7     89I 

10 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

03340374 

4  9   13 
4  8   12 

17   21   25 
16  20  24 

30  34  38 
28  32  37 

n 
12 

0414 
0792 

0453 
0828 

0492 
0864 

0531 
0899 

0569 
0934 

0607 
0969 

0645 
1004 

0682 
1038 

0719 
1072 

0755 
I106 

4  8   12 
4  7   II 
3   7   II 
3  7   10 

15   19  23 
IS  19  22 
14  18  21 
14   17   20 

27  31  35 
26  30  33 
25   28  32 

24  27  31 

13 

14 

1 139 
I46I 

1173 
1492 

1206 
1523 

1239 
1553 

1271 
1584 

1303 
1614 

1335 
1644 

1367 
1673 

1399 
1703 

1430 
1732 

3   7    10 
3   7   10 
369 
369 

13   16  20 
12   16   19 
12   IS   18 

12    IS    17 

23   26  30 
22   25   29 
21   24  28 
20  23   26 

IS 

I76I 

1790 

1818 

1847 

187s 

1903 

1931 

1959 

1987 

2014 

369 
3   S     8 

II  14  17 
II  14  16 

20  23   26 
19  22   25 

16 

17 

2041 
2304 

2068 
2330 

2095 
2355 

2122 
2380 

2148 
2405 

2I7S 
2430 

2201 
2455 

2227 
2480 

2253 
2504 

2279 
2529 

3   5     8 
358 
3   5     8 
257 

II  14  16 

10  13   IS 
10  13   15 
10  12   15 

19  22   24 
18  21   23 
18   20  23 
17   19   22 

18 
19 

2553 

2788 

2577 
2810 

2601 
2833 

2625 
2856 

2648 
2878 

2672 
2900 

2695 
2923 

2718 
2945 

2742 
2967 

276s 
2989 

2   5     7 
2  5     7 
2  4     7 
2  4     6 

9  12  14 
9   II   14 
9   II   13 
8   II   13 

16   19   21 
16   18   21 
16   18  20 

IS  17  19 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

2  4     6 

8   II   13 

15   17   19 

21 
22 
23 

3222 
3424 
3617 

3243 
3636 

3263 
3464 
3655 

3284 
3483 
3674 

3304 
3502 
3692 

3324 
3522 
3711 

3345 
3541 
3729 

3365 
3560 

3747 

3385 
3579 
3766 

3404 
3598 
3784 

246 
246 
246 

8    10    12 
8   10   12 
7     9   II 

14  16   18 
14  15   17 
13   15   17 

24 

11 

29 

3802 

3979 
41SO 

3820 
3997 
4166 

3838 
4014 
4183 

3856 
4031 
4200 

3874 
4048 
4216 

3892 
4065 
4232 

3909 
4082 
4249 

3927 
4099 
4265 

3945 
4116 
4281 

3962 
4133 
4298 

24s 
235 
23s 

7     9   II 
7     9   10 
7     8   10 

12   14  16 
12   14  IS 

II  13  15 

4314 
4472 
4624 

4330 
4639 

4346 
4502 
4654 

4362 
4518 
4669 

4378 
4533 
4683 

4393 
4548 
4698 

4409 
4564 
4713 

4425 
4579 
4728 

4440 
4594 
4742 

4456 
4609 

4757 

235 
235 
I  3     4 

689 
689 
679 

II  13  14 

II    12    14 
10    12     13 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

I  3     4 

679 

10  II  13 

31 
32 
33 

4914 
SOSi 
5185 

4928 
506s 
SI98 

4942 
5079 
5211 

4955 
5092 
5224 

4969 
5105 
5237 

4983 
5119 
5250 

4997 
5132 
5263 

5011 
S145 
5276 

5024 

nil 

5038 
5172 
5302 

I  3     4 
I   3     4 
I   3     4 

678 

5      1     I 
S     6     8 

10    II    12 
9    II    12 
9    10    12 

34 
U 

S3IS 
S44I 
5563 

5328 
5453 
5575 

5340 
5465 
5587 

5353 
5478 
5599 

5366 
5490 
5611 

5378 
5502 
5623 

5391 
5514 
5635 

5403 
5527 
5647 

5416 

5539 
5658 

542§ 
5551 
5670 

I  3     4 
124 
I   2     4 

5     6     8 
5     6     7 
567 

9  10  II 
9   10   II 
8   10   II 

11 
39 

40 

S682 
5798 
S9II 

6021 

5694 
5809 
5922 

5705 
5821 
5933 

5717 
5832 
5944 

5729 
5843 
5955 

5740 
5855 
5966 

5977 

5763 
5877 
5988 

5775 
5888 
5999 

5786 
5899 
6010 

I   2     3 
I   2     3 
123 

S     6     7 
567 

4     S     7 

8     9   10 
8     9   10 
8     9   10 

603 1 

6042 

6053 

6064 

60756085 

6096 

6107 

6117 

I   2     3 

4     5     6 

8     9   10 

41 
42 
43 

6128 
6232 
6335 

6138  6149 
6243  6253 
63456355 

6160 
6263 
6365 

6170 
6274 
6375 

6180  6191 
6284  6294 
6385  6395 

6201 
6304 
6405 

6212 
6314 
6415 

6222 
6325 
6425 

I   2     3 
I   2     3 
I   2     3 

4     S     6 
4     5     6 
4     5     6 

789 
789 
789 

44 

11 

643s 
6532 
6628 

6444 '6454 '6464 
6542;65Si|656i 
6637:66466656 

6474 
6571 
6665 

6484  6493 
6580  6590 
6675  6684 

6503 
6S99 
6693 

6513I6522 
660916618 
6702  6712 

I   2     3 
I   2     3 
I    2      3 

4     S     6 
4     S     6 
4     5     6 

789 
7     8     9 

778 

47 
48 
49 

6721 
6812 
6902 

673o'6739[6749 
6821  6830  6839 
691169206928 

6758 
6848 
6937 

6767  6776 
6857  6866 
6946695s 

6785 
6875 
6964 

6794^6803 
68846893 
69726981 

I   2     3 
I   2     3 
I   2     3 

4     5     5 
4     4     5 
4     4     5 

6     7     8 
678 
678 

50I6990 

6998  7007  7016 

70247033I7042 

7050 

7059  7067 

12     3I   3     4     5 

678 

468 


Logarithms 


7324 
7404 


54 
55 

56  7482 

57  7559 

58  7634 


59  7709 

60  7782 
6i  7853 


62  7924 

63  7993 

64  8062 


65 


75 


8129 


8195 
8261 
8325 


8388 
8451 
8513 


8573 
8633 
8692 


8751 


8808 
886s 
8921 


79  8976 

80  9031 

81  9085 


9138 
9191 
9243 


9294 


9345 
9395 
9445 


9494 
9542 
9590 


9638 
9685 
9731 


95i9777 


9823 
9868 
,9912 


)  o  1  I  I  3  I  3  I  4 

51  7076  7084  7093  7x01  7110 

52  7160  7168  7177  7185  7193 

53  7243  7251  7259  7267  7275 


6    I    7    1 


9    |i  2    3  1  4     5     6  i  7 


7332I7340 
741217419 


7348 
7427 


749017497  7505 
7566,7574  7582 
7642  7649  7657 


7716  7723 
77897796 
7860:7868 


7731 
7803 
7875 


7931  7938  7945 
8000  8007  8014 
8069  807s  8082 


8136  8142  8149 


8202  82098215 
8267  8274  8280 
8331  8338  8344 


8395  840118407 
8457  8463  8470 
8SI9|8525|853I 

8579  858518591 
8639  864s  8651 
8698  8704  8710 


8756  8762,8768 


8814  8820  8825 
8871  8876  8882 
8927  8932I8938 


8982  898718993 
9036  9042  9047 
9090  909619101 


9143  9I49|9IS4 

9196  9201  j9206 

9248925319258 


9299  93049309 


93509355  9360 
9400  9405  9410 
9450945519460 

9499'9504'9509 
9547,9552:9557 
9595  9600  9605 


9643  9647  9652 
9689  9694  9699 
9736,9741  i9745 


9782I9786J9791 


9827  9832  9836 
9872  98779881 
9917:9921:9926 


99i9956  9961 19965  9969  9974I9978I9983 


7118 
7202 
7284 


7356  7364  7372 
743517443  7451 


7126 
7210 
7292 


7SI3!7S20[7528 
7589  759717604 
7664  7672  7679 

7738^7745  7752 
7810  7818  7825 
7882,7889  7896 


7952  7959  7966 
8021  8028-8035 
8089  8096  8102 


8156  8162  8169 


8222  8228  8235 
8287  8293  8299 
8351  8357  8363 


8414  8420  8426 
8476  8482  8488 
8537  8S43]8549 

8597  8603  8609 
8657  8663  8669 
8716  8722  8727 


8774:877918785 


8831  8837  8842 
8887  8893  8899 
8943  89498954 


8998  9004  9009 
9053  9058  9063 
9106  9112  9117 


9159  916s  9170 
9212  9217  9222 
9263  9269  9274 


9315  93209325 


936s  9370 
9415  9420 
9465  9469 

95139518 
9562  9566 
9609  9614 


9657 
9703 
97SO 


9375 
9425 
9474 


9523 
9571 
9619 


9661  9666 
9708,9713 
97549759 


9795  9800980s 


9841 
9886 
9930 


9845  9850 
9890  9894 
9934,9939 


7135 
7218 
7300 


7143 
7226 
7308 


7152 
7235 
7316 


73807388I7396 
7459]7466j7474 


7536  75437551 
761217619  7627 
768617694  7701 


7760I7767 
78327839 
7903  7910 


7774 
7846 
7917 


797317980  7987 
8041  8048  8055 
8l09;8ll6;8i22 


8176  8182  8189 


8241  8248  8254 
8306  8312  8319 
837o;8376:8382 


8432  8439I8445 
8494  850018506 
8S55j856lJ8567 


861S  8621:8627 
8675  8681  8686 
8733;8739i8745 


8791  8797  8802 


8848,88548859 
8904  8910  891S 
8960  8965  8971 


9015  9020  9025 
906990749079 
912219128  9133 


9i75|9i8o  9186 
922719232  9238 
9279  92849289 


9330,9335,9340 


93809385  9390 
94309435  9440 
9479  9484  9489 


9528  9533  9538 
9576  9581  9586 
9624  9628  9633 


9671  9675  9680 
9717  9722  9727 
97639768  9773 


98099814  9818 


985498599863 
9899  9903  9908 
9943:9948,9952 


9987  999 1 '9996 


I    2 
I    2 


I  2 
I  2 
I    I 


I  I 
I  I 
I    I 


112 
112 
112 


I  I 
I  I 
I     I 


112 
112 
112 


I     I 
I     I 


112 
112 
112 


112 
112 


I  I 
0  I 
O    I 


0  I 
O  I 
0    I 


0  I 
O  I 
0    I 


O  I 
O  I 
0    I 


3  4  5 
3  4  5 
3     4     4 


3  4  4 
3  4  4 
3      4     4 


3  3  4 
3  3  4 
3     3     4 


3     4 


3  3  4 
3  3  4 
3     3     4 


3  4 
3  4 
3     4 


234 
234 
234 


233 
233 
233 


233 
233 
233 


233 
233 
233 


233 


3  3 
2  3 
2     3 


223 
223 
223 


2  2 
2  2 
2       2 


223 
223 
223 


5  6  7 
5  6  7 
567 


S  6  7 
5  6  6 
5     6  6 


S  6  6 
S  S  6 
S     5  6 


556 
5  5  6 
456 


4     5   5 


4  4  5 
4  4  5 
4     4  5 


4  4  5 
4  4  5 
4     4  5 


4     4  5 


4  4  5 
3  4  4 
3     4  4 


3  4  4 
3  4  4 
3     4  4 


3     4  4 


The  copyright  of  that  portion  of  the  above  table  which  gives  the  logarithms  of 
numbers  from  1000  to  2000  ia  the  property  of  Messrs.  Macmillan  and  Company, 
Li  Sited,  who,  however,  have  authorised  the  use  of  the  form  in  any  reprint  pub- 
lished for  educational  purposes. 


Logarithms  of  Trigonometric  Functions 


o       / 

log  sin 

d 

log  tan 

dc 

log  cot 

log  cos 

' 

S 

T 

°xS 

3011 

1 
3011 

0 . 0000 
0.0000 

0  90|        0 

SO             10 

1 

7.4637 

7.4637 

2.5363 

6.4637 

6.4637 

20 
30 
40 

7.7648 
7.9408 
8.06S8 

1760 

1250 

969 

7.7648 
7.9409 
8.0658 

1761 

1249  ! 

969 

1 

2.2352 
2.0591 
1.9342 

0.0000 
0.0000 
0.0000 

40       '■      20 

30    ;   30 
20    j   40 

6.4637 
6.4637 
6.4637 

6.4637 
6.4637 
6.4637 

SO 

I        0 

10 

8. 1627 
8.2419 
8.3088 

792 
669 
S8o 

8.1627 
8.2419 
8.3089 

792 
670 
S80 

1.8373 
I. 7581 
I.69II 

0.0000 
9.9999 
9.9999 

10 
0  89 

so 

11 

70 

6.4637I6.4638 
6.4637  6.4638 
6.4637  6.4638 

20 
30 
40 

8.3668 
8.4179 
8.4637 

Sii 
4S8 
413 

8.3669 
8.4181 
8.4638 

512 

4S7 
41S 

I. 6331 
1.5819 
1.5362 

9.9999 
9.9999 
9.9998 

40 
30 
20 

80 

90 

100 

6.4637 
6.4637 
6.4637 

6.4638 
6.4638 
6.4638 

SO 

2       0 
10 

8.5050 
8.5428 
8.5776 

378 
348 
321 

8.5053 
8.5431 
8.5779 

348 
322 

I . 4947 
1.4569 
I .4221 

9.9998 
9.9997 
9.9997 

10 
0  88 

SO 

no 
120 
130 

6.4637 
6.4636 
6 . 4636 

6 . 4639 
6.4639 
6.4639 

20 
30 
40 

8.6097 
8.6397 
8.6677 

300 
280 
263 

8.6101 
8.6401 
8.6682 

300 
281 
263 

1.3899 
1.3599 
I. 3318 

9 . 9996 
9 . 9996 
9.9995 

40 
30 
20 

140 
150 
160 

6.4636 
6.4636 
6.4636 

6.4640 
6 . 4640 
6 . 4640 

50 

3     0 

10 

8 . 6940 
8.7188 
8.7423 

248 
235 
222 

8 . 694s 
8.7194 
8.7429 

249 
235 
223 

1.305s 
1.2806 
I. 2571 

9.9995 
9.9994 
9.9993 

10 

0  87 
so 

170 
180 
190 

6.4635  6.4641 
6.463516.4641 
6.463s  6.4642 

20 

30 

40 

8.764s 
8.7857 
8.8059 

212 
202 
192 

8.7652 
8.7865 
8.8067 

213 
202 
194 

1.2348 
I. 2135 
I. 1933 

9.9993 
9.9992 
9.9991 

40 
30 
20 

200 
210 
220 

6.4635  6.4642 
6.463s  6.4643 
6. 4634 16. 4643 

so 

4     0 

10 

8.8251 
8.8436 
8.8613 

185 
177 
170 

8.8261 
8.8446 
8.8624 

185 
178 
171 

I. 1739 
I. 1554 
I. 1376 

9.9990 
9.9989 
9.9989 

10 

0  86 
SO 

230 
240 
250 

6.4634  6.4644 
6.4634  6.4644 
6.4633  6.4645 

20 
30 
40 

8.8783 
8.8946 
8.9104 

163 
158 

152 

8.8795 
8.8960 
8.9118 

165 
158 
154 

I . 1205 
I . 1040 
1.0882 

9.9988 
9.9987 
9.9986 

40 
30 
20 

260 
270 
280 

6.4633 
6.4633 
6.4632 

6 . 4646 
6.4646 
6.4647 

so 
5     0 

8.9256 
8.9403 

147 

8.9272 
8.9420 

148 

1.0728 
I .0580 

9.9985 
9.9983 

10 
0  8s 

290 
300 

6.4632 
6.4631 

6.4648 
6 . 4649 

1  log  cos    1    d     1  log  cot    1    dc    1  log  tan    |  log  sin    1  '     °  1           1               1             | 

143 

1  14.3 

2  28.6 

3  42.9 

142 

14.2 
28.4 
42.6 

138 

13.8 
27.6 
41.4 

137 

13. 
27. 
41. 

135 

7    13.5 
4    27.0 
1    40.5 

134 

13.4 
26.8 
40.2 

130 

13. ( 
26. ( 
39. ( 

129 

)    12.9 
)    25.8 
)    38.7 

127 

12.7 
25.4 
38.1 

125 

12.5 
25.0 
37.5 

123 

12.3 
24.6 
36.9 

122 

12.2 
24.4 
36.6 

119 

11. f 
23. J 
35.' 

117 

11.7 
23.4 
35.1 

115 

11.5 
23.0 
34.5 

114 

11.4 

22.8 
34.2 

4    57.2 
5j  71.5 
6    85.8 

56.8 
71.0 
85.2 

55.2 
69.0 

82.8 

54. 
68. 

82. 

S    54.0 
5    67.5 
2    81.0 

53.6 
67.0 
80.4 

52. ( 
65. ( 

78. ( 

)    51.6 
)    64.5 
)    77.4 

50.8 
63.5 
76.2 

50.0 
62.5 
75.0 

49.2 
61.5 
73.8 

48.8 
61.0 
73.2 

47. f 

46.8 
58.5 
70.2 

46.0 
57.5 
69.0 

45.6 
57.0 
68.4 

7  100.1 

8  114.4 
128.7 

99.4 
113.6 

127.8 

96.6 
110.4 
124.2 

95. 
109. 
123. 

9    94.5 
5  108.0 
3,121.5 

93.8 
107.2 
120.6 

91. ( 
104.  ( 
117. ( 

)    90.3 
)  103.2 
)ill6.1 

88.9 
101.6 
114.3 

87.5 
100.0 
112.5 

86.1 
98.4 
110.7 

85.4 
97.6 
109.8 

83. r 
95.5 
107.1 

81.9 
93.6 
105.3 

80.5 
92.0 
103.5 

79.8 
91.2 
102.6 

Formulas  for  using  Table  directly 

%  I  log  sin  jc  =  log  ic'  +  S                       °.o  f  log  cos  x  =  log  (90  - 
V  i  log  tan  X  =  log  x'  +  T                       °?  ■{  log  cot  x  =  log  (90  — 
H  [  log  cot  x  =  colog  a;'  +  CO  T             '^  [  log  tan  x  =  colog  (90 

xY  +  s 

xY  +  T 

-  xY  +  CO  T 

470 


Logarithms  of  Trigonometric  Functions 


log  sin 


log  tan 


dc 


log    cot     log    COS 


pp 


5     0 

10 


8.9403    142 


■  9545 


8.9682 
30j  8.9816 
40     8.9945 


SO 


SO 
7  o 


20 
30 
40 

50 

8  o 

10 

20 
30 
40 

so 

9  o 
10 


137 

134 
129 

125 


9.0070  122 

9.0192  119 

9.0311  IIS 

9.0426  113 

9.0S39  109 

9.0648  107 


9.07SS 
9.0859 
9.0961 

9.1060 
9-II57 
9.1252 

9.134s 
9.1436 
9.1525 

9.1612 
9.1697 
9.1781 

9.1863 
9.1943 
9.2022 

9.2100 
9.2176 
9.2251 


50  9.2324 
10  o  9.2397 


log  cos 


104 

102 

99 

97 
95 
93 

91 
89 
87 

85 
84 
82 

80 
79 
78 

76 
75 
73 


8.9420 
8.9563 

8.9701 
8.9836 
8.9966 

9 . 0093 
9.0216 
9.0336 

9-0453 
9.0567 
9.0678 

9.0786 
9.0891 
9 • 0995 

9.1096 
9.1194 
9.1291 

9.1385 
9.1478 
9.1569 

9.1658 
9-1745 
9.1831 

9.191S 
9.1907 
9.2078 

9.2158 
9.2236 
9.2313 

9.2389 
9.2463 


143 
138 

135 
130 
127 

123 
120 
117 

114 
III 
108 

105 
104 

lOI 

98 
97 
94 

93 
91 
89 

87 
86 
84 

82 
81 
80 

78 
77 
76 

74 


r.0580 
1.0437 

I . 0299 
I. 0164 
I . 0034 

0.9907 
0.9784 
0 . 9664 

0.9547 
0.9433 
0.9322 


9214 
9109 
9005 

8904 
8806 
8709 


0.8615 
0.8522 
0.8431 

0.8342 
0.8255 
0.8169 

0.8085 
0.8003 
0.7922 

0.7842 
0.7764 
0.7687 

0.7611 
0.7537 


log  cot      dc 


log  tan 


9.9983 
9.9982 


o  8s 

50 


9.9981  40 
9.9980130 
9.9979  20 


9-9977 
9-9976 
9-9975 

9-9973 
9-9972 
9.9971 

9 . 9969 
9.9968 
9.9966 

9.9964 
9 . 9963 
9.9961 

9.9959 
9.9958 
9.9956 

9.9954 
9-9952 
9-9950 

9.9948 
9.9946 
9 . 9944 


10 

o  84 
50 

40 
30 
20 

10 
o  83 

50 

40 
30 
20 

10 
0  82 

50 

40 
30 
20 

10 
o  81 

50 


9.9942140 
9.9940  30 
9.9938  20 


9  -  9936 
9.9934 


10 
o  80 


log  sin 


94    I 

9.4' 
18.8! 
28. 2| 

37.6 
47.0 
56.4] 

65.8! 
75. 2| 
84.6 


93 

9.3 
18.6 
27.9 

37.2 
46.5 
55.8 

65.1 
74.4 
83.7 


91 

9.1 
18.2 
27.3 

36.4 
45.5 
54.6 

63.7 
72.8 
81.9 


89    I  87    I  86 

8.9  8.7  8.6 
17.8,  17.41  17.2 
26. 7i  26.1    25.8 


35.6!  34.8 
44. 5j  43.5 
53. 4|  52.2 

60.9 


62.3 
71.2 
80. 1:  78.3 


34.4 
43.0 
51.6 


85 

8.5 
17.0 
15.5 

34.0 
42.5 
51. Ol 

59.5! 
68.01 
76.51 


log*' 

logx' 

colog  x! 


68.8 

77.4 
Formulas  for 
log  sin  X  —  S 
log  tan  X  —  T 
log  cot  «  —  CO  T 


84 

8.4 
16.8 
25.2 

33.6 
42.0 
50 

58.8 
67.2 
75.6 


82 

8.2 
16.4 
24.6 


81 


16. 2I 
2.34 


79 

7.9 
15.8 
23.7 


"3 

II. 3 
22.6 
33.9 

45-2 
56. 5 
67.8 

79.1 
90.4 
101.7I   99.9 

108 

10.8 

21.6 


33 

44. 
55 
66, 

77.7 
88.8 


32.4 
43.2 
54.0 
64.8 

75-6 
86.4 
97-2 


107 

10.7 
21.4 
32.1 
42.8 
53-5 
64-2 

74. 91 
85-6 
96.31 


104 

10.4 

20.8 

31.2 

102 

10.2 
20.4 
30.6 

41.6 

52.0 

62.4 

40.8 

72.8 
83-2 

93-6 

l\:t 

91.0 

109 

10.9 

21.8 

32.7 

43-6 
54-5 
65-4 

76.3 
87.2 
98.1 

105 

10.5 
21.0 
31-5 
42.0 
52. 5 
63.0 

73-5 
84.0 
94-5 
lOI 

10. 1 
20.2 
30-3 

40.4 
50.5 
60.6 

70.7 
80.8 
90.9 


99 

9.9 
19.8 
29.7 
39-6 
49.5 


98      97 

9.8 
19.6 


29.4 

39-2 
49.0 


659-458.8 

7 


95 

9.71   9-5 
19.4  19.0 

29.1  28.5 
38.838.0 
48.S'47.5 

58.2  57-0 


69.3  68.6I67.966.5 
79.2  78.4i77.6  76.0 
89. i|88. 2187.3185. 5 


78  1  77 

7.81  7.7 
15.6  15.4 
23.4:23. 1 


76  I  75  I  74 

7.6  7.5   7.4 
15.215.014.8 

22.822.522.2 


30.4'30.029.6 


.645. 

.252. 

o'crk 


51.8 
59.2 


43. 

51.1 
58.4 


hi 

471 


32.8    32.4!  31.631.230.8 

41.0;  40.5    39.5,39.0  38.538.0,37.537.036.5 

49.21  48.6]  47.446.8|46.245.6|45.044.4 

57.4    56.7!  55.3'54.6  53.9  53 

65. 6i  64.8,  63. 262.4161. 660. „, „„.. 

73.8    72.9!  71.l|70.2|69.368.4!67.566.665.7 

Table  inversely 
log  (90  —  xY  —  log  cos  *  —  S 
log  (90  —  xY  =  log  cot  x  —  T 

colog  (90  —  x)'  =  log  tan  re  —  co  T 


73 

7.3 
14.6 
21.9 

29.2 


Logarithms  of  Trigonometric  Functions 


o       / 

log  sin 

d 

log  tan 

dc 

log  cot 

log  cos 

d 

PP 

10     O 

9.2397 

71 

9.2463 

73 

0.7537 

9.9934 

3 

0 

80 

73 

71 

10 

9.2468 

70 

9.2536 

73 

0.7464 

9.9931 

2 

50 

I 
2 

7.3 
14.6 

7.1 
14.2 

20 

9.2538 

68 

9.2609 

71 

0.7391 

9.9929 

2 

40 

3 

21.9 

21.3 

30 

9.2606 

68 

9.2680 

70 

0.7320 

9.9927 

3 

30 

4 
5 

29.2 
36.5 

28.4 

35-5 

40 

9.2674 

66 

9.2750 

69 

0.7250 

9.9924 

2 

20 

SO 

9.2740 

66 

9.2819 

68 

0.7I8I 

9.9922 

3 

10 

6 

43-8 

42.6 

II     0 

9.2806 

64 

9.2887 

66 

0.7II3 

9-9919 

2 

0 

79 

7 

51-I 

4M 

.„ 

9.2870 

64 

9-2953 

67 

0.7047 

9.9917 

3 

50 

8 
9 

58. 4 
65-7 

56.8 
63-9 

20 

9.2934 

63 

9.3020 

65 

0.6980 

9.9914 

2 

40 

70 

7.0 
14.0 

5^ 

6.9 

13.8 

30 

9.2997 

61 

9.3085 

64 

0.6915 

9.9912 

3 

30 

I 

40 

9.30S8 

61 

9.3149 

63 

0.6851 

9  -  9909 

2 

20 

2 

so 

9.3119 

60 

9.3212 

63 

0.6788 

9.9907 

3 

10 

3 

21.0 

20.7 

12     0 

9.3179 

59 

9.3275 

61 

0.6725 

9.9904 

3 

0 

78 

4 

28.0 

27.6 

10 

9.3238 

58 

9-3336 

61 

0.6664 

9.9901 

2 

50 

5 
6 

35.0 
42.0 

34. 5 
41.4 

20 

9.3296 

57 

9-3397 

61 

0.6603 

9.9899 

3 

40 

7 
8 

49.0 
S6.o 
63.0 

48.3 
55.2 
62.1 

30 

9.33S3 

^1 

9.3458 

59 

0.6542 

9.9896 

3 

30 

40 

9.3410 

56 

9.3517 

59 

0.6483 

9-9893 

3 

20 

9 

so 

9.3466 

55 

9.3576 

58 

0.6424 

9.9890 

3 

10 

68 

67 

13    0 

9.3521 

54 

9.3634 

57 

0.6366 

9.9887 

3 

0 

77 

I 

6.8 

6.7 

10 

9-3575 

S4 

9-3691 

57 

0.6309 

9.9884 

3 

50 

2 
3 

13.6 
20.4 

13.4 
20.1 

20 

9.3629 

53 

9.3748 

56 

0.6252 

9.9881 

3 

40 

4 
5 
6 

27.2 
34-0 
40.8 

26  8 

30 

9.3682 

52 

9.3804 

55 

0.6196 

9.9878 

3 

30 

33^5 
40.2 

40 

9.3734 

52 

9.3859 

55 

0.6I4I 

9.9875 

3 

20 

so 

9.3786 

51 

9.3914 

54 

0.6086 

9-9872 

3 

10 

7 

47.6 

46.9 

14    0 

9.3837 

50 

9.3968 

S3 

0.6032 

9.9869 

3 

0 

76 

8 

54-4 

53-6 

10 

9.3887 

SO 

9.4021 

53 

0.5979 

9.9866 

3 

50 

9 

161.2 

60.3 

20 

9.3937 

49 

9-4074 

S3 

0.5926 

9.9863 

4 

40 

I 

66 

6.6 

65 

6.5 
13.0 
19.5 

30 

9.3986 

49 

9.4127 

SI 

0.5873 

9.9859 

3 

30 

2 

13-2 
19.8 

40 

9.403s 

48 

9.4178 

52 

0.5822 

9.9856 

3 

20 

3 

so 

9.4083 

47 

9.4230 

SI 

0.5770 

9.9853 

4 

10 

4 

26.4 

26.0 

IS    0 

9.4130 

9.4281 

O.S7I9 

9.9849 

0 

75 

5 

33-0 

32. 5 

H-' 

6 

7 

39-6 
46.2 

39.0 

45.5 

log  cos 

d 

log  cot 

dc 

log  tan 

log  sin 

d 

/ 

0 

8 

52.8 

52.0 

959.4 

1S8.S 

~ 

64 

63 

6 

I 

60 

59 

5^ 

57 

s^ 

55 

54 

53 

52 

^ 

51   1 

50 

48 

47 

I 

6.4 

6.3 

6 

.1 

6.0 

5-9 

5-8 

5.7 

5.6 

5.5 

5.4 

5.3 

5 

2 

S.I 

5-0 

4.8 

4-7 

2 

12.8 

12.6 

12 

.2 

12.0 

II. 8 

II. 6 

II. 4 

II. 2 

[I.O 

10.8 

10.6 

10 

4 

10.2  ] 

0.0 

9.6 

9.4 

3 

19.2 

18.9 

18 

.3 

18.0 

17.7 

17-4 

17. 1 

16.8 

[6.5 

16.2 

15.9 

IS 

6 

IS. 3  1 

5-0  ] 

4-4 

14. 1 

• 

4 

25.6 

25.2 

24 

•  4 

24.0 

23-6 

23.2 

22.8 

22.4. 

22.0 

21.6 

21.2 

20 

8 

20.45 

'0.0  1 

9.2 

18.8 

5 

32.0 

31.5 

30 

.5 

30.0 

29-5 

29.0 

28. s 

28.0  . 

27.5 

27.0 

26.5 

26 

0 

25.5  5 

55.05 

4.0 

23 -5 

6 

38.4 

37.8 

36 

.6 

36.0 

35-4 

34.8 

34.2 

33.6. 

J3.0 

32.4 

31.8 

31 

2 

30.6 : 

50.0  5 

8.8 

28.2 

7 

44.8 

44.1 

42 

.7 

42.0 

41.3 

40.6 

39.9 

39.2  . 

58. 5 

37.8 

37.1 

36 

4 

35.7  : 

55.0: 

3.6 

32.9 

8 

SI. 2 

so. 4 

48 

.8 

48.0 

47-2 

46.4 

45.6 

44-8. 

^4.0 

43-f 

42.4 

41 

6 

40.8  z 

to.o: 

8.4 

37.6 

9 

57.6 

56.7 

S4 

.9 

54-0 

S3-i|52.2 

SI. 3 

50.4^ 

^9.5 

48.6 

47.7 

46 

8 

45.9^ 

[5-o\a 

t3.2 

42.3 

472 


Logarithms  of  Trigonometric  Functions 


°     '  i  log  sin 

d 

log  tan 

dc 

log  cot 

log  cos 

d 

1 

PP 

IS    0    9.4130 

47 

9.4281 

so 

0.5719 

9.9849 

3 

0  75 

10    9.4177 

46 

9.4331 

50 

0.5669 

9.9846 

3 

50 

50      49 

20j    9.4223 

46 

9.4381 

49 

0.5619 

9.9843 

4 

40 

I 

5.0    4.9 

30     9.4269 

4S 

9.4430 

49 

0.5570 

9.9839 

3 

30 

2 

lo.o    9-8 

40 

9.4314 

45 

9.4479 

48 

0.5521 

9.9836 

4 

20 

3 

15.0:14.7 

SO 

9.43S9 

44 

9.4527 

48 

O.S473 

9.9832 

4 

10 

4 

20.0  19.6 

16    0 

9.4403 

44 

9. 4575 

47 

0.5425 

9.9828 

3 

0  74 

5 

25.0  24. 5 

10 

9.4447 

44 

9.4622 

47 

0.5378 

9.982s 

4 

SO 

6 

30.029.4 

20 

9.4491 

42 

9.4669 

47 

0.5331 

9.9821 

4 

40         1 

7 

35.034.3 

30 

9.4S33 

43 

9.4716 

46 

0.5284 

9.9817 

3 

30 

8 

40.039.2 

40 

9.4S76 

42 

9.4762 

46 

0.5238 

9.9814 

4 

20 

9 

45.044.1 

50 

9.4618 

41 

9.4808 

45 

0.5192 

9.9810 

4 

10 

17     0 

9.46S9 

41 

9.4853 

45 

0.5147 

9.9806 

4 

0  73 

10 

9.4700 

41 

9.4898 

45 

0.5102 

9.9802 

4 

50 

I 

4»„ 

4.8 

47 

4.7 

20 

9.4741 

40 

9.4943 

44 

O.S057 

9.9798 

4 

40 

2 

9.6 

9.4 

30 

9.4781 

40 

9.4987 

44 

O.SO13 

9.9794 

4 

30 

3 

14.4 

14. 1 

40 

9.4821 

40 

9.S031 

44 

0.4969 

9.9790 

4 

20 

4 

19.2 

18.8 

SO 

9.4861 

39 

9.S075 

43 

0.4925 

9.9786 

4 

10 

5 

24.0 

23.  5 

18    0 

9.4900 

39 

9.5118 

43 

0.4882 

9.9782 

4 

0  72 

6 

28.8 

28.2 

10 

9.4939 

38 

9.S161 

42 

0.4839 

9.9778 

4 

50 

7 

33.6 

32.9 

20 

9.4977 

38 

9.5203 

42 

0.4797 

9.9774 

4 

40 

8 

38.4 

37.6 

30 

9.S01S 

37 

9.5245 

42 

0.475s 

9.9770 

5 

30 

9 

43.2 

42.3 

40 

9.S0S2 

38 

9.5287 

42 

0.4713 

9.9765 

4 

20 

SO 

9.S090 

36 

9.5329 

41 

0.4671 

9.9761 

4 

10 

19    0 

9.S126  !  37 

9.5370 

41 

0.4630 

9  9757 

5 

0 

46 

45 

10 

9.5163 

36 

9.5411 

40 

0.4589 

9.9752 

4 

50  71 

I 
2 

4.6 
9.2 

4.5 
9.0 

20 

9.S199 

36 

9.5451 

40 

0.4549 

9.9748 

S 

40 

3 

13.8 

13. 5 

30 

9.S23S 

35 

9.5491 

40 

0.4509 

9  9743 

4 

30 

40 

9.S270 

36 

9.5531 

40 

0.4469 

9.9739 

S 

20 

4 
5 

18.4118.0 
23.022.5 

SO 

9.S306 

9.5571 

40 

0.4429 

9.9734 

4 

10 

6 

27.6 

27.0 

20    0 

9.S34I 

35 

9.5611 

0.4389 

9.9730 

0  70 

7 

3^« 

31. 5 

8 

36.8 

36.0 

log  cos 

d 

log  cot 

dc 

log  tan 

log  sin 

d 

/      0 

9 

41.4140.5 

44 

43 

42 

41 

40 

39 

38 

37 

3^ 

35 

I 

4-4 

4.3 

4.2 

I 

4.1 

4.0 

3.9 

3.8 

I 

3. 

7    3.6 

35 

2 

8.8 

8.6 

ix 

2 

8.2 

8.0 

7.8 

7.6 

2 

7. 

4    7.2 

7.0 

3 

13-2 

12.9 

3 

12.3 

12.0 

II. 7 

II. 4 

3 

II. 

I  10.8 

10. S 

4 

17.6 

17.2 

16.8 

4 

16.4 

16.0 

15.6 

IS. 2 

4 

^^ 

8  14.4 

14.0 

5 

22.0 

21. 5 

21.0 

s 

20.5 

20.0 

19.5 

19.0 

5 

18 

5  18.0 

17.5 

6 

26.4 

25.8 

25.2 

6 

24.6 

24.0 

23.4 

22.8 

6 

22 

2121. 6 

.,.0 

7 

30.8 

30.1 

29.4 
33.6 

7 

28.7 

28.0 

27.3 

26.6 

7 

25 

9|2S.2 

24. 5 

8 

3S.2 

34.4 

8 

32.8 

32.0 

31.2 

30.4 

8 

29 

628.8 

28.0 

9 

39.6 

!38.7 

37.8 

9 

36.9 

36.0 

35. 1 

34.2 

9 

33 

3132.4 

31.5 

473 


Logarithms  of  Trigonometric  Functions 


O    1 

log  sin 

d 

log  tan 

dc 

log  cot 

log  COS 

d 

PP 

20  0 

10 

9.S34I 
9.S37S 

34 
34 

9.5611 
9.5650 

39 
39 

0.4389 
0.4350 

9.9730 
9.972s 

S 
4 

0  70 

50 

4 

0.4 
0.8 
1.2 

20 
30 
40 

9 . 5409 
9 .  5443 
9-5477 

34 
34 
33 

9.5689 
0.5727 
9.5766 

38 
39 
38 

0.43II 
0.4273 
0.4234 

9.9721 
9.9716 
9.97II 

5 
5 
5 

40 
30 
20 

I 
2 
3 

50 

21   0 

10 

9.5510 
9.5543 
9.5576 

33 
33 
33 

9.5804 
9.5842 
9.5879 

38 
37 
38 

0.4196 

0.4158 
0.4I2I 

9.9706 
9.9702 
9.9697 

4 
5 
S 

10 
0  69 

SO 

4 
5 
6 

1.6 
2.0 
2.4 

20 
30 

40 

9.5609 
9.5641 
9.5673 

32 
32 
31 

9.5917 
9.5954 
9.5991 

37 
37 
37 

0.4083 
0 . 4046 
0.4009 

9.9692 
9.9687 
9.9682 

5 

5 
5 

40 
30 
20 

7 
8 
9 

2.8 
3.2 
3.6 

SO 

22   0 

10 

9.5704 
9.5736 
9.5767 

32 
31 
31 

9.6028 
9.6064 
9.6100 

36 
36 
36 

0.3972 
0.3936 

0.3900 

9.9677 
9.9672 
9.9667 

5 
5 
6 

10 
0  68 

50 

5 

20 
30 
40 

9.5828 
9.5859 

30 
31 
30 

9.6136 
9.6172 
9.6208 

3f 
36 

35 

0.3864 
0.3828 
0.3792 

9.9661 

9.9656 
9.9651 

5 
5 
5 

40 
30 
20 

I 
2 
3 

0.5 

I.O 

i.S 

50 

23  0 

10 

20 
30 
40 

9.5889 
9.5919 
9.5948 

9.5978 
9.6007 
9.6036 

30 
29 
30 

29 
29 
29 

9.6243 
9.6279 
9.6314 

9.6348 
9.6383 
9.6417 

36 
35 
34 

35 
34 
35 

0.3757 
0.3721 
0.3686 

0.3652 
0.3617 
0.3583 

9 . 9646 
9.9640 
9.963s 

9.9629 
9.9624 
9.9618 

6 
5 
6 

5 
6 
5 

10 
0  67 
SO 

40 
30 
20 

4 
5 
6 

7 
8 
9 

2 . 0 

2.5 

3.0 

3.5 
4.0 

4-5 

SO 
24  0 

10 

9.6065 
9 . 6093 
9.6121 

28 
28 
28 

9.6452 
9.6486 
9.6520 

34 
34 
33 

0.3548 
0.3514 
0.3480 

9.9613 
9.9607 
9.9602 

6 

I 

10 

0  66 
50 

I 

6 

0.6 

20 
30 
40 

9.6149 
9.6177 
9.6205 

28 
28 
27 

9.6553 
9.6587 
9.6620 

34 
33 
34 

0.3447 
0.3413 

0.3380 

9.9596 
9.9590 
9.9584 

6 
6 
5 

40 
30 
20 

2 

3 

4 

I 

7 
8 

1 . 2 
1.8 

2.4 

3° 
3.6 

4.2 
4.8 

SO 
25  0 

9.6232 
9.6259 

27 

9.6654 
9.6687 

33 

0.3346 
0.3313 

9.9579 
9.9573 

6 

10 
0  65 

log  COS 

d 

log  cot 

dc 

log  tan 

log  sin 

d 

/  0 

9 

5.4 

I 

2 

3 

39 

3.9 

7.8 
II. 7 

38 

3.8 
7.6 

II. 4 

37 

3.7 

7.4 

II. I 

3^ 

3.6 

7.2 
10.8 

35 

3.5 

7.0 

10. 5 

34 

3.4 

6.8 

10.2 

33 

3.3 
6.6 
9.9 

32 

3-2 

31 

I: 

9-, 

30 

c  3.0 
2  6.0 
J  9.0 

29 

2.9 

i:? 

28 

2.8 
5.6 
8.4 

27 

2.7 
5.4 
8.1 

4 
5 
6 

IS. 6 
19.5 
23.4 

15.2 
19.0 
22.8 

14.8 
18.5 
22.2 

14.4 
18.0 
21.6 

14.0 
17.5 
21.0 

13.6 
17.0 
20.4 

li:s 
19.8 

12.8 
16.0 
19.2 

12.^ 
15.. 
18. ( 

\  12.0 
;  15.0 
3 18.0 

II. 6 
14. 5 
17.4 

II. 2 
14.0 
16.8 

10.8 

7 
8 
9 

27.3 
31.2 
35.1 

26.6 
30.4 
34-2 

25.9 
29.6 
33.3 

25.2 
28.8 
32.4 

24. S 
28.0 
31. 5 

23.8 
27.2 
30.6 

23.1 
26.4 
29.7 

22.4 
25.6 
28.8 

21.' 
24.5 
27. ( 

1  21.0 

i  24.0 
)  27.0 

20.3 
23.2 
26.1 

19.6  18.9 
22.421.6 
25.2  24.3 

474 


Logarithms  of  Trigonometric  Functions 


o   / 

log  sin 

d 

log  tan 

dc 

log  cot 

log  COS 

d 

PP 

25  o 

9.6259 

27 

9.6687 

33 

0.3313 

9.9S73 

6 

0 

65 

10 

9.6286 

27 

9.6720 

32 

0.3280 

9.9S67 

6 

50 

20 

9.6313 

27 

9.6752 

33 

0.3248 

9.9S61 

6 

40 

30 

9.6340 

26 

9.678s 

32 

0.32IS 

9.9SSS 

6 

30 

t> 

40 

9.6366 

26 

9.6817 

33 

0.3183 

9.9S49 

6 

20 

I 
2 

0.6 

I  2 

50 

9.6392 

26 

9.6850 

32 

0.3ISO 

9.9S43 

6 

10 

3 

I'.S 

26  0 

9.6418 

26 

9.6882 

32 

0.3II8 

9.9537 

7 

0 

64 

10 

9.6444 

26 

9.6914 

32 

0.3086 

9.9530 

6 

50 

4 

s 

2.4 
3.0 

•  20 

9.6470 

25 

9.6946 

31 

0.3054 

9.9524 

6 

40 

6 

3.6 

30 

9.649s 

26 

9.6977 

32 

0.3023 

9.9518 

6 

30 

40 

9.6521 

25 

9 . 7009 

31 

0.2991 

9. 9512 

7 

20 

7 
8 

4.2 
4.8 

50 

9.6546 

24 

9.7040 

32 

0.2960 

9.950s 

6 

10 

9 

5.4 

27  0 

9.6570 

25 

9.7072 

31 

0.2928 

9 ■ 9499 

7 

0 

63 

1 

10 

9.659s 

25 

9.7103 

31 

0.2897 

9.9492 

6 

so 

7 
0.7 

20 

9.6620 

24 

9.7134 

31 

0.2866 

9 . 9486 

7 

40 

I 

30 

9 . 6644 

24 

9.7165 

31 

0.2835 

9.9479 

6 

30 

2 

1.4 

40 

9.6668 

24 

9.7196 

30 

0.2804 

0.9473 

7 

20 

3 

2. 1 

SO 

9.6692 

24 

9.7226 

31 

0.2774 

9.9466 

7 

10 

4 

2.8 

28  0 

9.6716 

24 

9.7257 

30 

0.2743 

9. 9459 

6 

0 

62 

5 

3.5 

10 

9.6740 

23 

9.7287 

30 

0.2713 

9.9453 

7 

SO 

6 

4.2 

20 

9.6763 

24 

9.7317 

31 

0.2683 

9 . 9446 

7 

40 

7 

4.9 

30 

9.6787 

23 

9.7348 

30 

0.2652 

9.9439 

7 

30 

8 

5.6 

40;  9.6810 

23 

9.7378 

30 

0.2622 

9.9432 

7 

20 

9 

6.3 

SO  9.6833 

23 

9.7408 

30 

0.2592 

9.942s 

7 

10 

1 

29  0 

9.6856 

22 

9.7438 

29 

0.2562 

9.9418 

7 

0 

61 

8 

10 

9.6878 

23 

9.7467 

30 

0.2S33 

9.9411 

7 

SO 

I 

0.8 
1 .6 

20 

9.6901 

22 

9.7497 

29 

0.2503 

9.9404 

7 

40 

3 

2.4 

30 

9.6923 

23 

9.7526 

30 

0.2474 

9.9397 

7 

30 

40 

9 . 6946 

22 

9.7SS6 

29 

0.2444 

9.9390 

7 

20 

4 
5 

3.2 

4.0 

SO 

9.6968 

9.7S8S 

29 

0.241S 

9.9383 

8 

10 

6 

4.8 

30  0 

9.6990 

22 

9.7614 

0.2386 

9.937s 

0 

60 

• 

7 

5.6 

8 

6.4 

1  log  COS 

d 

log  cot 

dc 

log  tan 

log  sin 

d 

/ 

0 

9 

7.2 
22 

33 

32 

31 

30 

29 

27 

26 

25 

»4 

33 

I 

3.3 

3.2 

3.1 

I 

3.C 

2.9 

2.' 

J    2.6 

2.5 

I 

2.4 

2.3 

2.2 

2 

6.6 

6.4 

6.2 

2 

6.C 

S.8 

s.^ 

X    5.2 

5.0 

2 

4.8 

4.6 

ff 

3 

9.9 

9.6 

9.3 

^ 

9.C 

8.7 

8. 

I  7.8 

7.5 

3 

7.2 

6.9 

6.6 

4 

13.2 

12.8  I 

2.4 

4 

12. c 

>  II. 6 

10. 

5  10.4 

10. 0 

4 

9.6 

9.2 

8.8 

5 

16.5 

16.0  I 

S.S 

5 

iS.c 

>  14-5 

13. 

5  13.0 

12. 5 

5 

12.0 

II. s 

II. 0 

6 

19.8 

19.2  I 

8.6 

t 

18. c 

>  17.4 

16. 

2  iS-6 

15.0 

6 

14.4 

13.8 

13.2 

7 

23.1 

22.4  2 
25.6  2 

1.7 

21. c 

)  20.3 

18. 

?I8.2 

17. 5 

7 

16.8 

16.] 

15.4 

8 

26.4 

4.8 

i 

24. c 

)  23.2 

21. 

f)j20.8 

20.0 

8 

19.2 

18. A 

^  17.6 

9 

29.7 

28.8^2 

7.9 

S 

)  27. c 

)  26.1 

24. 

3123.4 

22.5 

9 

21.6 

20.- 

119. 8 

475 


Logarithms  of  Trigonometric  Functions 


o   . 

log  sin 

d 

log  tan 

dc 

log  cot 

log  cos 

d 

PP 

30  0 

9.6990 

22 

9.7614 

30 

6.2386 

9.9375 

7 

0  60 

10 

9.7012 

21 

9.7644 

29 

0.2356 

9.9368 

7 

50 

20 

9.7033 

22 

9.7673 

28 

0.2327 

9.936X 

8 

40 

7 

30 

9.7055 

21 

9.7701 

29 

0.2299 

9.9353 

7 

30 

X 

0.7 

40 

9.7076 

21 

9.7730 

29 

0.2270 

9 . 9346 

8 

20 

2 

3 

1.4 
2.  I 

SO 

9.7097 

21 

9.7759 

29 

0.2241 

9.9338 

7 

10 

31  0 

9.7118 

21 

9.7788 

28 

0.22X2 

9.9331 

8 

0  59 

4 

2.8 

10 

9.7139 

21 

9.7816 

29 

0.2184 

9.9323 

8 

50 

5 
6 

3.5 

4.2 

20 

9.7160 

21 

9.7845 

28 

0.2I5S 

9.9315 

7 

40 

30 

9.7181 

20 

9.7873 

29 

0.2127 

9.9308 

8 

30 

7 

4.9 

40 

9.7201 

21 

9.7902 

28 

0.2098 

9.9300 

8 

20 

8 
9 

i:t 

SO 

9.7222 

20 

9.7930 

28 

0.2070 

9.9292 

8 

xo 

32  0 

9.7242 

20 

9.7958 

28 

0.2042 

9.9284 

8 

0  58 

10 

9.7262 

20 

9.7986 

28 

0.2014 

9.9276 

8 

SO 

I 

8 

0.8 

20 

9.7282 

20 

9.8014 

28 

0.1986 

9.9268 

8 

40 

2 

1.6 

30 

9.7302 

20 

9.8042 

28 

0.1958 

9.9260 

8 

30 

3 

2.4 

40 

9.7322 

20 

9.8070 

27 

0.1930 

9.9252 

8 

20 

4 

3.2 
4.0 

SO 

p. 7342 

19 

9.8097 

28 

0.1903 

9.9244 

8 

xo 

5 

33  0 

9.7361 

19 

9.812s 

28 

O.X875 

9.9236 

8 

0  57 

6 

4.8 

10 

9.7380 

20 

9.8153 

27 

0.1847 

9.9228 

9 

50 

7 

5.6 

20 

9.7400 

19 

9.8180 

28 

0. 1820 

9.92x9 

8 

40 

8 

6.4 

30 

9.7419 

19 

9.8208 

27 

0.1792 

9.92II 

8 

30 

9 

1  7.2 

40 

9.7438 

19 

9.823s 

28 

0.176s 

9.9203 

9 

20 

9 

0.9 

SO 

P. 7457 

19 

9.8263 

27 

0.1737 

9.9194 

8 

xo 

X 

34  0 

9.7476 

18 

9.8290 

27 

0.X7I0 

9.9x86 

9 

0  56 

2 

X.8 

10 

9.7494 

19 

9.8317 

27 

0.1683 

9.9177 

8 

50 

3 

2.7 

20 

9.7SI3 

18 

9.8344 

27 

0.1656 

9.9169 

9 

40 

4 

3.6 

30 

9.7531 

19 

9.8371 

27 

0. 1629 

9.9x60 

9 

30 

5 

4-5 

40 

9.7SSO 

18 

9.8398 

27 

0.1602 

9.9I5I 

9 

20 

6 

5.4 

SO 

9.7568 

9.8425 

0.1575 

9.9142 

XO 

7 

6.3 

35  0 

9.7586 

18 

9.8452 

27 

0.XS48 

9.9134 

8 

0  55 

8 
9 

7.2 
8.x 

log  cos 

d 

log  cot 

dc 

log  tan 

log  sin 

d 

/  0 

30 

29 

28 

27 

22 

21 

20 

1  '9 

18 

I 

3.( 

)   2.< 

)   2.8 

[  2.7 

2.2 

2.1 

I 

2.0 

1.9 

1.8 

2 

6.( 

>  s.| 

i    5.6 

2  5.4 

4.4 

4.2 

2 

4.0 

3.8 

3.6 

3 

9.< 

7     8.4 

' 

5  8.1 

6.6 

6.3 

3 

6.0 

5.7 

5.4 

4 

12. < 

5  II. ( 

D  II. 2 

\  X0.8 

8.8 

8.4 

4 

8.0 

7.6 

7.2 

S 

IS.< 

J  14. 

5  14.0 

;  135 

II. 0 

10.5 

5 

10. 0 

95 

9.0 

6 

18. < 

3  17.. 

\  16.8 

( 

3  16.2 

13.2 

12.6 

6 

12.0 

II. 4  I 

0.8 

7 

21. 

3  20. 

J  19.6 

7  18.9 

15.4 

14.7 

7 

14.0 

13.3  I 

2.6 

8 

24. 

3  23. 

2  22.4 

5  21.6 

17.6 

x6.8 

8 

x6.c 

IS. 2  I 

4.4 

9 

27. 

3  26. 

I  2S.2 

( 

P  24.3 

19.8 

18.9 

9 

x8.c 

!i7.i  I 

6.2 

476 


Logarithms  of  Trigonometric  Functions 


o   t 

log  sin 

d 

log  tan 

dc 

log  cot 

log  cos 

d 

PP 

35  0 

9.7S86 

18 

9.84S2 

27 

0.1548 

9.9134 

9 

0  55 

10 

9.7604 

18 

9.8479 

27 

0. 1521 

9.9125 

9 

SO 

20 

9.7622 

18 

9.8506 

27 

0.1494 

9.9II6 

9 

40 

I 

9 

0.9 

30 

9.7640 

17 

9.8533 

26 

0.1467 

9.9107 

9  30 

2 

1.8 

40 

9.7657 

18 

9.8559 

27 

O.I44I 

9.9098 

9  '20 

3 

2.7 

50 

9.767s 

17 

9.8586 

27 

0.I4I4 

9.9089 

9  !io 

4 

3.6 

36  0 

9.7692 

18 

9.8613 

26 

0.1387 

9.9080 

10  1  0  54 

S 

4.5 

10 

9.7710 

17 

9.8639 

27 

O.I36I 

9.9070 

9  SO 

6 

5.4 

20 

9.7727 

17 

9.8666 

26 

O.X334 

9.9061 

9 

40 

7 

6.3 

30 

9.7744 

17 

9.8692 

26  i  0.1308  1 

9.9052 

10 

30 

8 

7.2 

40 

9.7761 

17 

9.8718 

27 

0. 1282 

9.9042 

9 

20 

9 

8.1 

SO 

9.7778 

17 

9.8745 

26 

0.I2SS 

9.9033 

10 

10 

1 

37  0 

9.779s 

16 

9.8771 

26 

0.1229 

9.9023 

9 

0  53 

10 

10 

9.7811 

17 

9.8797 

27 

0.1203 

9.9014 

10 

50 

I 

I.O 

20 

9.7828 

16 

9.8824 

26 

O.II76 

9.9004 

9 

40 

2 
3 

2 . 0 
3.0 

30 

9.7844 

17 

9.8850 

26 

O.II5O 

9.899s 

10 

30 

40 

9.7861 

16 

9.8876 

26 

O.II24 

9.8985 

10 

20 

4 
5 

4.0 
S.o 

SO 

9.7877 

16 

9.8902 

26 

0.1098 

9.897s 

10 

10 

6 

6.0 

38  0 

9.7893 

17 

9.8928 

26 

0. 1072 

9.8965 

10 

0  52 

10 

9.7910 

16 

9.8954 

26 

0.1046 

9.895s 

10 

SO 

7 
8 

7.0 
8.0 

20 

9.7926 

IS 

9.8980 

26 

0. 1020 

9.894s 

10 

40 

9 

9.0 

30 

9.7941 

16 

9.9006 

26 

0.0994 

9.8935 

10 

30 

40 

9.79S7 

16 

9.9032 

26 

0.0968 

9.892s 

10 

20 

SO 

9.7973 

16 

9.9058 

26 

0 . 0942 

9.8915 

10 

10 

I 

I.I 

39  o|  9.7989 

IS 

9.9084 

26 

0.0916 

9.8905 

10 

0  51 

2 

2.2 

10  9.8004 

16 

9.9110 

25 

0.0890 

9.8895 

II 

SO 

3 

3.3 

20 

9.8020 

IS 

9.913s 

26 

0.086s 

9.8884 

10 

40 

4 

4.4 

30 

9.803s 

IS 

9.9161 

26 

0.0839 

9.8874 

10 

30 

5 

55 

40 

9.80SO 

16 

9.9187 

25 

0.0813 

9.8864 

II 

20 

6 

6.6 

SO 

9.8066 

IS 

9.9212 

26 

0.0788 

9.8853 

10 

10 

7 

7.7 

40  0 

9.8081 

9.9238 

0.0762 

9.8843 

0  50 

8 

8.8 

9 

9-9 

log  COS 

d 

log  cot 

dc 

log  tan 

log  sin 

d 

' 

0 

IS 

27 

26 

25 

18 

17 

16 

I 

2.7 

2.6 

I 

2.5 

i.i 

5  1.7 

I 

I. 

3  1.5 

2 

5.4 

S.2 

2 

5.0 

3.( 

J  3.4 

2 

3. 

2  3.0 

3 

8.1 

7.8 

3 

7.5 

s./ 

\    S.I 

3 

4. 

5  4.5 

4 

10.8 

10.4 

4 

10. 0 

n.' 

J  6.8 

4 

6. 

\    6.0 

5 

13. 5 

13.0 

S 

12.5 

9.< 

)  8.5 

5 

8. 

3  7.5 

6 

16.2 

IS. 6 

6 

15.0 

10. { 

J  10.2 

6 

9.< 

3  9.0 

7 

18.9 

18.2 

7 

17.5 

12. ( 

3  II. 9 

7 

II. 

2  10.5 

8 

21.6 

20.8 

8 

20.0 

14.^ 

\  13. 6 

8 

12. 

3  12.0 

9 

24.3 

23.4 

9 

22.5 

16. 

2;i5.3 

9 

14. 

%  13. S 

477 


Logarithms  of  Trigonometric  Functions 


o   , 

log  sin 

d 

log  tan 

dc 

log  cot 

log  cos 

d 

PP 

40  0 

9.8081 

IS 

9.9238 

26 

0.0762 

9.8843 

II 

0  50 

10 

9 . 8096 

IS 

9.9264 

2S 

0.0736 

9.8832 

II 

so 

20 

9.8111 

14 

9.9289 

26 

0.07 1 1 

9.8821 

II 

40 

10 

30 

9.812s 

IS 

9. 93 IS 

26 

0.0685 

9.8810 

10 

30 

I 

I.O 

40 

9.8140 

15 

9.9341 

25 

0.0659 

9.8800 

II 

20 

2 
3 

2.0 
3.0 

SO 

9.815s 

14 

9.9366 

26 

0.0634 

9.8789 

II 

10 

4.0 
5.0 
6.0 

41  0 

9.8169 

IS 

9.9392 

25 

0.0608 

9.8778 

II 

0  49 

4 
5 
6 

10 

9.8184 

14 

9.9417 

26 

0.0583 

9.8767 

II 

SO 

20 

9.8198 

IS 

9.9443 

25 

0.0557 

9.8756 

II 

40 

7 

7.0 

30 

9.8213 

14 

9.9468 

26 

0.0532 

9.874s 

12 

30 

8 

8.0 

40 

9.8227 

14 

9.9494 

25 

0.0506 

9.8733 

II 

20 

9 

9.0 

SO 

9.8241 

14 

9.9SI9 

25 

0.0481 

9.8722 

II 

10 

42  0 

9.82SS 

14 

9 -9544 

26 

0.0456 

9.87II 

12 

0  48 

10 

9.8269 

14 

9.9S70 

25 

0.0430 

9.8699 

II 

SO 

II 

1. 1 

20 

9.8283 

14 

9.9S9S 

26 

0 . 0405 

9.8688 

12 

40 

I 

30 

9.8297 

14 

9.9621 

25 

0.0379 

9.8676 

II 

30 

2 

2.2 

40 

9.8311 

13 

9 . 9646 

25 

0.0354 

9.866s 

12 

20 

3 

3.3 

so 

9.8324 

14 

9.9671 

26 

0.0329 

9.8653 

12 

10 

4 
S 
5 

4.4 

43  0 

9.8338 

13 

9-9697 

25 

0.0303 

9.8641 

12 

0  47 

10 

9.83SI 

14 

9.9722 

25 

0.0278 

9.8629 

II 

50 

7 

7.7 

20 

9.836s 

13 

9.9747 

25 

0.0253 

0.8618 

12 

40 

8 

8.8 

30 

9.8378 

13 

9.9772 

26 

0.0228 

9.8606 

12 

30 

9 

9.9 

40 

9.8391 

14 

9.9798 

25 

0.0202 

9.8594 

12 

20 

SO 

9.840s 

13 

9.9823 

25 

0.0177 

9.8582 

13 

10 

44  0 

9.8418 

13 

9.9848 

26 

0.0152 

9.8569 

12 

0  46 

12 

10 

9.8431 

13 

9.9874 

25 

0.0126 

9.8557 

12 

50 

I 
2 

1.2 

20 

9.8444 

13 

9.9899 

25 

O.OIOI 

9.8S4S 

13 

40 

3 

30 

9.84S7 

12 

9.9924 

25 

0.0076 

0.8532 

12 

30 

4 
S 

4.8 
6.0 

40 

9.8469 

13 

9.9949 

26 

0.0051 

9.8520 

13 

20 

SO 

9.8482 

9.997s 

25 

0.0025 

9.8507 

12 

10 

6 

7.2 

45  0 

9.849s 

13 

0.0000 

0.0000 

9.8495 

0  45 

7 
8 
9 

10.8 

log  cos 

d 

log  cot 

dc 

log  tan 

log  sin 

d 

/   0 

13 

26 

25 

IS 

14 

I 

^2.6 

I 

2.S 

I 

1.5      I 

1.4 

c  1.3 

2 

S.2 

2 

S.o 

2 

3.0      2 

2.8     : 

J  2.6 

3 

7.8 

3 

7.S 

3 

4.5      3 

4.2    : 

J  3.9 

4 

10.4 

4 

10. 0 

4 

6.0      4 

5.6 

i     S.2 

S 

13.0 

S 

12. s 

5 

7.5      5 

7.0 

>  6.5 

6 

IS. 6 

6 

iS.o 

6 

9.0      6 

8.4    < 

>  7.8 

7 

18.2 

7 

17. s 

7 

10.5      7 

9.8 

J    9.1 

8 

20.8 

8 

20.0 

8 

12.0      8 

II. 2    i 

5  10.4 

9 

23-4 

9 

22.5 

9 

13.5      9 

12.6    ( 

)  II. 7 

478 


Natural  Trigonometric  Functions 


Deg. 

Radians 

n  sin 

n  CSC 

n  tan 

n  cot 

n  sec 

n  cos 

o 

0.0000 

.000 

.000 

1. 000 

1. 00 

1.5708 

90 

I 

3 

0.0175 
0.0349 
0.0524 

.017 
.035 
.052 

57.3 
28.7 
19. 1 

.017 
.035 
.052 

57.3 
28.6 
19. 1 

1. 000 

1. 001 
1. 001 

1. 00 
.999 
.999 

1.5533 
1.5359 
1. 5184 

89 
88 
87 

4 
S 
6 

0.0698 
0.0873 
0.1047 

.070 
.087 
.105 

14-3 
II. 5 
9.57 

.070 
.087 
.105 

14-3 
II. 4 
9.51 

1.002 
1.004 
1.006 

.998 
.996 
.995 

I. 5010 
I. 4661 

86 

84 

7 
8 
9 

0. 1222 
0.1396 
0.IS7I 

.  122 
.139 
.156 

8.21 
7.19 
6.39 

.123 
.141 
.158 

8.14 
7.12 
6.31 

1.008 
1. 010 
1. 012 

.993 
.990 
.988 

1.4486 
I. 4312 
I. 4137 

83 
82 
81 

lO 

0.1745 

.174 

5.76 

.176 

5.67 

1. 015 

.985 

1.3963 

80 

II 

12 

13 

0.1920 
0.2094 
0.2269 

.191 
.208 
.225 

5.24 
4.81 
4-45 

.194 
.213 
.231 

5.14 
4-70 
4-33 

1. 019 
1.022 
1.026 

.982 
.978 
.974 

1.3788 
I. 3614 
1.3439 

79 

78 

77 

14 
IS 
i6 

0.2443 
0.2618 
0.2793 

.242 
.259 
.276 

3.63 

.287 

4.01 
3.73 
3.49 

1. 031 
1.035 
1.040 

.970 
.966 
.961 

1.326s 
1.3090 
I. 2915 

76 

75 
74 

17 

i8 

•   19 

0.2967 
0.3142 
0.3316 

.292 
.309 
.326 

3.42 
3.24 
3.07 

.306 
.325 
.344 

3.27 
3.08 
2.90 

1.046 
1.051 
1.058 

.956 
.951 
.946 

I. 2741 
1.2566 
1.2392 

73 
72 
71 

20 

0.3491 

.342 

2.92 

.364 

2.75 

1.064 

.940 

I. 2217 

70 

21 
22 
23 

0.366s 
0.3840 
0.4014 

.358 
.375 
.391 

2.79 

2.67 
2.56 

.384 
.404 
•  424 

2.61 
2.48 
2.36 

1. 07 1 
1.079 
1.086 

.934 
.927 
.921 

I . 2043 
I. 1868 
I. 1694 

69 
68 
67 

24 
25 
26 

0.4189 
0.4363 
0.4538 

.407 
.423 
.438 

2.46 
2.37 
2.28 

■.til 

.488 

2.25 
2.14 
2.05 

1.095 
1. 103 
1. 113 

.906 
.899 

1.1519 
I. 1345 
1.1170 

66 
64 

27 
28 
29 

0.4712 
0.4887 
0.5061 

.454 
.469 
.485 

2.20 
2.13 
2.06 

.510 
.532 

.554 

1.96 

1.88 
1.80 

1. 122 
1. 133 

1. 143 

.891 
.883 
.875 

I . 0996 
I. 0821 
I . 0647 

62 
61 

30 

0.5236 

.500 

2.00 

.577 

1.73 

1. 155 

.866 

1.0472 

60 

31 
32 

33 

0.5411 
0.5585 
0.5760 

.515 
.530 
.545 

1.94 
1.89 
1.84 

.601 
.625 
.649 

1.66 
1.60 
1.54 

1. 167 
1. 179 
1. 192 

.857 
.848 
.839 

1.0297 
I. 0123 
0.9948 

59 
58 
57 

34 

11 

O.S934 
0.6109 
0.6283 

.559 

■.lit 

1.79 
1.74 
1.70 

.675 
.700 
.727 

1.48 
1.43 
1.38 

1.206 
1. 221 
1.236 

.829 
.819 
.809 

0.9774 
0.9599 
0.9425 

56 
55 
54 

37 
38 
39 

0.6458 
0.6632 
0.6807 

.602 
.616 
.629 

1.66 
1.62 
1.59 

■.Hi 

.810 

1:11 

1.23 

1.252 
1.269 
1.287 

.799 
.788 
.777 

0.9250 
0.9076 
0.8901 

53 

52 
SI 

40 

0.6981 

.643 

1.56 

.839 

1. 19 

1.305 

.766 

0.8727 

SO 

41 
42 
43 

0.7156 
0.7330 
0.750s 

.656 
.669 
.682 

1.52 
1.49 
1.47 

.869 
.900 
.933 

1. 15 
I. II 
1.07 

1*346 
1.367 

.755 
.743 
.731 

0.8552 
0.8378 
0.8203 

49 
48 
47 

44 
45 

0.7679 
0.7854 

.695 
.707 

1.44 
1. 41 

.966 
1. 00 

1.04 
1. 00 

1.390 
1-414 

.719 
.707 

0.8029 
0.7854 

46 

45 

n  cos 

n  sec 

n  cot 

n  tan 

n  CSC 

nsin 

Radians 

Deg. 

479 


Antiloqarithms 


io|    I    I2i3i4isi6    rris    I9I123I4    5    6,7    8     ol 

•so 

.51 

•52 

.53 

3162 

3170  3177  3184 

3I92|3I99  3206 

3214 

3221 

3228 

112 

3     4     4 

5     6     7 

3236 
3311 
3388 

3243  3251  3258 
3319  3327  3334 
339634043412 

326632733281 
3342:33503357 
342034283436 

3289 
3365 
3443 

3296 
3373 
3451 

3304 
3381 
3459 

12       2 
12       2 
12       2 

3     4     5 
3     4     5 
3     4     5 

5     6     7 
5     6     7 
667 

.54 

■M 
il 

.59 
.60 

3467 
3548 
3631 

3475  3483  3491 
35563565  3573 
3639  3648  3656 

349935083516 
3581  35893597 
3664I3673  3681 

3524 
3606 
3690 

3532 
3614 
3698 

3540 
3622 
3707 

12       2 
12       2 
I    2       3 

3     4     5 
3     4     5 
3     4     5 

667 
677 
6     7     8 

371S 
3802 
3890 

3724  3733  3741 
3811  3819  3828 
389939083917 

3750  3758  3767 
383738463855 
3926  3936  3945 

377637843793 
38643873  3882 
39543963  3972 

I    2       3 
I    2       3 
I    2       3 

3  4     5 

4  4     5 
4     5      5 

678 
6     7     8 
678 

3981 

399039994009 

4018  4027  4036 

4046  4055  4064 

I    2       3 

456 

6     7     8 

.61 
.62 
.63 

4074 
4169 
4266 

4083  4093 
4i78i4i88 
42764285 

4102 
4198 
4295 

4111 
4207 
4305 

4121I4130 
421714227 
43154325 

4140  4150  4159 
42364246J4256 
4335!4345i4355 

I    2       3 
I    2       3 
I    2       3 

456 
4     5     6 
456 

7     8     9 
789 
789 

it 

436s 
4467 
4S7I 

4375 
4477 
4581 

4385 
4487 
4592 

4395 
4498 
4603 

4406 
4508 
4613 

4416  4426 
45194529 
4624  4634 

4436  4446'4457 
4539  4S50;456o 
464546564667 

I    2       3 
I    2       3 
I    2       3 

456 
4     5     6 
456 

789 
7      8     9 
7     9   10 

ii 
.69 

4677 
4786 
4898 

4688 
4797 
4909 

4699 
4808 
4920 

4710 
4819 
4932 

4721 
4831 
4943 

4732 
4842 
4955 

4742 
4853 
4966 

4753 
4864 
4977 

4764 
4875 
4989 

4775 
4887 
5000 

I    2       3 
I    2       3 
I    2       3 

4  5      7 
467 

5  6     7 

8     9  10 
8     9   10 
8     9   10 

.70 

5012 

5023 

5035 

5047 

S058 

SO70 

5082 

5093 

5105 

SI17 

1    2       4 

S     6     7 

8     9   II 

.71 
.72 
.73 

5129 

5248 
5370 

5140 
5260 
5383 

5152 
5272 
5395 

5164 
5284 
5408 

S176 
5297 
5420 

5188 
5309 
5433 

5200 
5321 

5445 

5212 
5333 
5458 

5224 
5346 
5470 

5236 
5358 
5483 

I    2       4 
I    2       4 
I    3       4 

5     6     7 
567 
568 

8  10  II 

9  10  II 
9   10  II 

■74 
•75 
.76 

5495 
S623 
5754 

5508 
5636 
5768 

5521 
5649 
5781 

5534 
5662 
5794 

5546 
567s 
5808 

5559 
5689 
5821 

5572 
5702 
5834 

5585 
5715 
5848 

5598 
5728 
5861 

5610 

5741 
5875 

I  3     4 
I   3     4 
I   3     4 

568 
5      7      8 
5     7      8 

9   10   12 
9   10   12 
9   II    12 

.79 

5888 
6026 
6166 

5902 
6039 
6180 

5916 
6053 
6194 

5929 
6067 
6209 

5943 
6081 
6223 

5957 
6095 
6237 

5970 
6109 
6252 

5984 
6124 
6266 

5998 
6138 
6281 

6012 
6152 
629s 

I   3     4 
I   3     4 
I  3     4 

5      7      8 
678 
679 

10   II    12 
10   II    13 
10   II    t3 

.80 

6310 

6324 

6339 

6353 

6368 

6383 

6397 

6412 

6427 

6442 

I   3     4 

679 

10   12   13 

.81 
.82 
.83 

6457 
6607 
6761 

6471 
6622 
6776 

6486 
6637 
6792 

6501 
6653 
6808 

6516 
6668 
6823 

6531 
6683 
6839 

6546 
6699 
6855 

6561 
6714 
6871 

6577 
6730 
6887 

6592 
6745 
6902 

235 
23s 
235 

689 
689 
689 

II    12    14 
II    12    14 
II    13   14 

1 

6918 
7079 
7244 

6934 
7096 
7261 

6950 
7112 
7278 

6966 
7129 
7295 

6982 
714s 
7311 

6998 
7161 
7328 

7015 
7178 

7345 

7031 
7194 
7362 

7047 
7211 
7379 

7063 
7228 
7396 

2  3      5 
235 
235 

6  8   10 

7  8   10 
7     8   10 

11  13   15 

12  13   IS 
12    13    15 

:89 

7413 
7586 
7762 

7430 
7603 
7780 

7447 
7621 
7798 

7464 
7638 
7816 

7482 
7656 
7834 

74997516 
7674  7691 
7852  7870 

75347551 
7709  7727 
78897907 

7568 
7745 
7925 

2  3     5 

245 
245 

7     9   10 
7     9   II 
7     9   II 

12    14    16 

12  14    16 

13  14    16 

.90 

7943 

7962 

7980 

7998 

8017 

8035  8054 

8072  8091 

8110 

2  4     6 

7     9   II 

13    15    17 

.91 
.92 
.93 

8128 
8318 
8511 

8147 
8337 
8531 

8166 
8356 
8551 

8185 
8375 
8570 

8204 
8395 
8590 

8222'824I 
84148433 
8610,8630 

8260 
8453 
8650 

8279 
8472 
8670 

8299 
8492 
8690 

2  4     6 
2  4     6 

246 

8     9   II 
8   10   12 
8   10  12 

13  15    17 

14  IS  17 

14  16    18 

.94 

8710 
8913 
9120 

8730 
8933 
9141 

8750 
8954 
9162 

8770 
8974 
9183 

8790 
899s 
9204 

8810  8831 
901619036 
92269247 

8851 
9057 
9268 

8872 
9078 
9290 

8892 
9099 
9311 

2  4     6 
2  4     6 
2  4     6 

8   10  12 
8   10  12 
8   II    13 

14  16    18 

15  17  19 
IS  17  19 

-.11 

99 

9333 
9550 
9772 

9354 
9572 
9795 

9376 
9594 
0817 

9397 
9616 
9840 

9419 
9638 
9863 

9441  9462 
9661I9683 
98869908 

9484 
9705 
9931 

9506 
9727 
9954 

9528 
9750 
9977 

247 
247 
2  5     7 

9   II    13 
9   II    13 
9    II    14 

15  17   20 

16  18  20 
16  18  20 

480 


Antilogarithms 


lo|il2|3l4lS     16|7     1819    |i  23 14     5      61    7     89I 

•00 

1000 

1002 

1005  1007 

1009 

1012 

1014 

1016 

1019 

I02I 

001 

III 

222 

•01 
•02 
03 

1023 

1047 
1072 

1026 
1050 
1074 

1028  1030 
1052  I0S4 
10761079 

1033 
1057 
1 08 1 

1035 
1059 
1084 

1038 
1062 
1086 

1040 
1064 
1089 

1042 
1067 
1091 

104s 
1069 
1094 

001 
0    0       I 
0   0       1 

III 
III 
III 

222 
222 
2       2       2 

•04 

■11 

1096 
1122 
1148 

1099 
1125 
1151 

1102 
1127 
IIS3 

II04 
II30 
II56 

1107 
1132 
1159 

1109 
1135 
I161 

1112 
1138 
1 164 

1114 
1 140 
1167 

I117 
1 143 
1169 

III9 
1 146 
II72 

0    I       I 
0    1       I 
0    I       I 

112 
112 
112 

2  2  2 
2  2  2 
2       2       2 

■11 

•09 

II7S 
1202 
1230 

I178 
1205 
1233 

I180 
1208 
1236 

II83 

I2II 
1239 

1186 
1213 
1242 

1 189 
1216 
1245 

I191 
1219 
1247 

1194 
1222 
1250 

1197 
1225 
1253 

1 199 
1227 
1256 

0    I       I 
0    1       I 

oil 

112 
112 
112 

2  2  2 
223 
223 

•10 

•II 
•12 
13 

1259 

1288 
1318 
1349 

1262 

1265 

1268 

1271 

1274 

1276 

1279 

1282 

1285 

oil 

112 

223 

1291 
1321 
1352 

1294 
1324 
1355 

1297 
1327 
1358 

1300 
1330 
1361 

1303  1306 
1334  1337 
1365  1368 

1309 
1340 
1371 

I3X. 

1343 
1374 

I315 
1346 
1377 

oil 

0    1       I 
0    I       I 

12       2 
12       2 
12       2 

223 
223 
233 

•14 

■.\l 

•19 

•20 

•21 
•22 

•23 

1380 
1413 
1445 

1384 
1416 
1449 

1387 
1419 
1452 

1390 
1422 
1455 

1393 
1426 
1459 

1396  1400 
1429  1432 
1462  1466 

1403 
1435 
1469 

1406 
1439 
1472 

1409 
1442 
1476 

0    1        I 
0    I       I 
0    1       I 

12       2 
12       2 
12       2 

233 
233 
233 

1479 
1514 
1549 

1483 
IS17 
1552 

i486 
1521 
1556 

1489 
1524 
1560 

1493 
1528 
1563 

1496 
1531 
1567 

1500 
1535 
1570 

1503 
1538 

1574 

1507 
1542 
1578 

I5IO 
1545 
I581 

01     I 
oil 
01     I 

12       2 
12       2 
12       2 

233 
233 

3     3     3 

1585 

1589 

1626 
1663 
1702 

1592 

1629 
1667 
1706 

1596 

1633 
167I 
I7IO 

1600 

1637 
1675 
1714 

1603 

1641 
1679 
1718 

1607 

x6ii 

I6I4 

1618 

oil 

12       2 

3     3     3 

1622 
1660 
1698 

1644 
1683 
1722 

1648 
1687 
1726 

1652 
1690 
1730 

1656 
1694 
1734 

oil 

0    1       I 

oil 

2       2       2 
2       2       2 
2       2       2 

3  3  3 
3  3  3 
3     3      4 

•24 

■11 

1738 
1778 
1820 

1742 
1782 
1824 

1746 
1786 
1828 

1750 
I79I 
1832 

1754 
1795 
1837 

1758 
1799 
1841 

1762 
1803 
1845 

1766 
1807 
1849 

1770 
I8II 
1854 

1774 
I816 
1858 

oil 
oil 

0    1       I 

2       2       2 
2       2       2 
223 

3  3  4 
3  3  4 
3     3     4 

11 

•29 

1862 
19OS 
1950 

1866 
1910 
1954 

1871 
1914 
1959 

1875 
I9I9 
1963 

1879 
1923 
1968 

1884 
1928 
1972 

1888 
1932 
1977 

1892 
1936 
1982 

1897 
I94I 
1986 

I9OI 

1945 
1991 

oil 

0    I       I 
0    1       I 

223 
2       2      3 
223 

3     3     4 

3  4  4 
3     4     4 

30 

1995 

2000 

2004 

2009 

2014 

2018 

2023 

2028 

2032 

2037 

oil 

223 

3     4     4 

31 
•32 

•33 

•34 

■11 

2042 
2089 
2138 

2046 
2094 
2143 

2193 

2244 
2296 

2051 
2099 
2148 

2056 
2104 

2153 

2061 
2109 
2158 

2065 
2113 
2163 

2070 
2118 
2168 

2075 
2123 
2173 

2080 
2128 
2178 

2084 
2133 
2183 

oil 

0    1       I 

oil 

223 
223 
223 

3  4  4 
3  4  4 
3     4     4 

2188 
2239 
2291 

2198 
2249 
2301 

2203 

2254 

2307 

2208 
2259 
2312 

2213 
2265 
2317 

2218 
2270 
2323 

2223 
2275 
2328 

2228 
2280 
2333 

2234 
2286 
2339 

233 
233 
233 

4  4  5 
4  4  5 
4     4     5 

11 

39 

2344 
2399 
24SS 

2350 
2404 
2460 

2355  2360 
241012415 
246612472 

2366 
2421 
2477 

2371 
2427 
2483 

2377 
2432 
2489 

2382 
2438 
2495 

2388 
2443 
2500 

2393 
2449 
2506 

233 
233 
233 

4  4  5 
4  4  5 
4     5     5 

.40 

2512 

2518 

2S23i2529 

2535 

2541 

2547 

2553 

2559 

2564 

234 

4     5     5 

•41 
.42 
•43 

2570 
2630 
2692 

2698 

2582I2588 
2642 1 2649 
2704  2710 

2594 
2655 
2716 

2600 
2661 
2723 

2606 
2667 
2729 

2612 
2673 
2735 

2618 
2679 
2742 

2624 
2685 
2748 

234 
234 
3     3     4 

4  5  5 
4  S  6 
4     S     6 

•44 

■■tl 

2754 
2818 
2884 

2761 
2825 
2891 

276712773 
2831  2838 
2897  2904 

2780 
2844 
2911 

2786 
2851 
2917 

2793 
2858 
2924 

2799 
2864 
2931 

2805 
2871 
2938 

2812 
2877 
2944 

112 

3     3     4 
3     3     4 
3     3     4 

4  S  6 
1     I     t 

.47  2951 
.48;3020 
.4913090 

295812965  2972 
30273034  3041 
3097I3105I3112 

2979  2985  2992 
3048  30553062 
3119131263133 

2999 
3069 
3141 

30063013 
30763083 
3I48I3I55 

112 
112 
112 

3     3     4 
3     4     4 

3     4     4 

I  i  t 

5     6     6 

31 


481 


INDEX 

(The  numbers  refer  to  the  pages) 


Abscissa,  26 

Absolute  value  of  complex  num- 
ber, 356 
Addition  formulas  for  sine  and 
cosine,  286-288 
for  tangent,  289 
Additive    properties    of    graphs, 

42,  273-276 
Algebraic  function,  13,  14 

scale,  3,  342 
Alternating   current   curves,  362 
et  seq. 
represented    by    complex 
numbers,  368 
Amplitude  of   complex  number, 
356 
of  S.  H.  M.,  322 
of  sinusoid,  113 
of  uniform  circular   motion, 

99 
of  wave,  326 
Angle,  96 

depression,  124 
direction,  100 
eccentric,  140 
elevation,  124 
epoch,  322,  326 
phase,  322,  326 
that  one  line  makes  with  an- 
other, 293 
vectorial,  100 
Angular  magnitude,  96 
units  of  measure,  97 
velocity,  99,  322 
Anti-logarithm,  232 


Approximation  formulas,  193 
Approximations,  successive,  180 
Argument  of  function,  10 

of  complex  number,  356 
Arithmetical  mean,  198 

progression,  198-201 

triangle,  188 
Asymptotes  of  hyperbola,  154, 157 
Auxiliary  circles,  140 
Axes  of  ellipse,  138 

of  hyperbola,  157 

Binomial    coefficients,    graphical 
representation   of,    196, 
197 
theorem,  189  et  seq. 
Briggs,  Henry,  216 

system  of  logarithms,  223 

Cartesian  coordinates,  26 

Cassinian  ovals,  387,  389 

Catenary,  274 

Change  of  base,  242,  243 

of  unit,  62,  73  et  seq.,  263 

Characteristic,  229,  230 

Circle    and    circular    functions, 
Chap.  III. 

Circle,  dipolar,  389 

equation  of,  94,  95 
sine  and  cosine,  120,  121 
tangent  to,  422,  427 
through  three  points,  433 

Circular    functions,    100    et    seq. 
graphical  computation  of; 
103,  111 


483 


484 


INDEX 


Circular  fundamental    relations, 
107,  282-297 
motion,  99 
Cologarithm,  233 
Combinations,    183,    186,    Chap, 

VI. 
Common  logarithms,  223 
Complementary  angles,  111,  114 
Completing  square,  457 
Complex    numbers,    Chap.    XI, 
341  et  seq. 
defined,  348 
laws  of,  350 
polar  form,  358 
typical  form,  348 
Composite  angles,  functions  of, 

290-293 
Composition  of  two  S.  H,  M.'s, 

324 
Compound  harmonic  motion,  334 
interest,  205,  211 
law,  256 
Computers  rules,  309 
Conditional  equations,  132,  300- 

315 
Conies,  413,  415 
con-focal,  443 

sections,  Chap.  XIII,  398  et 
seq. 
Conjugate  axis,  157 

complex  numbers,  352 
hyperbola,  158 
Connecting  rod  motion,  338 
Constants  and  variables,  13 
Continuous  function,  10 

compounding  of  interest,  256 
Coordinate  paper,  26,  119,  267- 

274 
Coordinates,  Chap.  II,  26  et  seq. 
Cartesian,  26 
orthogonal,  119 
polar,  118,  433 


Coordinates,  rectangular,  26,  27 
et  seq. 

relation  of  polaF  and  rectan- 
gular, 13 i,  433 
Cosine,  100 

curve,  113,  120 

law,  301 
Crest  of  sinusoid,  113 
Cubical  parabola,  50 
Cubic  equation,  177  et  seq. 
''Cut  and  Try,"  135 
Cycloid,  390 

Damped  vibrations,  276 
Damping  factor,  277 
Decreasing  function,  58 

geometrical  series,  206 
DeMoivres  theorem,  373 
Descartes,  Rene,  26 
Diameter  of  any  curve,  442 
of  ellipse,  442 
of  parabola,  419 
Direction  of  ellipse,  401,  413 
of  hyperbola,  406,  413 
of  parabola,  411,  413 
Discontinuous  function,  11,  31,  55 
Distance  of  point  from  line,  425 
Distributive   law   of   multiplica- 
tion, 189,  351 
general,  456 
Double  angle,  functions  of,  295 
scale,  5-8,  20,  21,  245-265 
of  algebraic  functions,  20 
of    logarithmic    functions, 
245-265 

"e,"  220,  223,  238,  257 

Eccentric  angle,  140 

Eccentricity  of  earth's  orbit,  402 
of  ellipse,  400 
of  hyperbola,  406 
of  parabola,  411,  413 


INDEX 


485 


Ellipse,   137    et   seq.;  398  et  seq. 
Chaps.  IV  and  XIII. 

axes  of,  138 

construction,  142,  143 

directrices,  401,  413 

eccentricity,  400 

focal  radii,  398,  429 

foci,  398 

latus  rectum,  403 

parametric  equation,  140 

polar  equation,  409 

shear  of,  436  '    ' 

symmetrical  equation  of,  138 

tangent  to,  428 

vertices,  138 
Elliptic  motion,  325,  382 
Empirical  curves,  261,  274 

formulas,  71 
Envelope,  421 

Epicycloid  and  epitrochoid,  393 
Epoch  angle,  322,  326,  330 
Even  function,  115 
Exponential  curves,  237-244 

equation,  211,  219 

function  Chap.  VIII,  214  et 
seq. 
defined,  219,  221,  223 
compared  with  power,  265 
Exponents,  definition  of,  460 

irrational,  222 

laws  of,  460 

Factorial  number,  183 
Factoring,  453-455 

fundamental  theorem  in,  457 
Factor  theorem,  163 
Family  of  curves,  74 

of  lines,  420 
Focal  radii  and  foci,  386 
of  ellipse,  398,  429 
of  hyperbola,  404 

radius  of  parabola,  412 


Frequency  of  S.  H.  M.,  323 

of  sinusoidal  wave,  329 

uniform  circular  motion,  99 
Function,  of  a  function,  91 

periodic,  30,  113,  360 

power,  46  et  seq.,  265 

rational,  14,  162 

S.  H.  M.,  327 

trigonometric,  100 
Functions,  9,  10 

algebraic,  13,  14 

circular.    Chap.    Ill,    94     et 
seq.,  100 

continuous,  10 

discontinuous,  11,  31,  55 

even  and  odd,  115 

explicit  and  implicit,  139 

exponential,   219,   221,   223, 
265 

increasing    and    decreasing, 
58,  152 

integral,  14,  162 

General  equation  of  second  de- 
gree, 437-440 
Geometrical  mean,  202 

progression,  202  et  seq. 
Graphical  computation,  15  et  seq. 
of  integral  powers,  19 
of  logarithms,  217 
of  product,  16,  388 
of  quotient,  17,  87,  88 
of  reciprocals,  89 
of  sq.  roots,  18,  21 
of  squares,  18,  21,  87 
solution  of  cubic,  177 
simultaneous   equations, 
174  et  seq. 
Graph  of  arithmetical  series,  200 
of  binomial  coefficients,  196, 

197 
of  complex  number,  349 


486 


INDEX 


Graph  of  cycloid,  392 

of  ellipse,  142,  143 

of  equation,  37 

of  functions  of  mutilple  an- 
gles, 298,  299 

of  geometrical  series,  207- 
210 

of  hyperbola,  153-156 

of  hyperbolic  functions,  274 

of  logarithmic  and  exponen- 
tial curves,  216,  217, 
237  et  seq. 

of  parabolic  arc,  420 

of  power  function,  46,  48,  59, 
73,  86 

of  sinusoid,  112 

of  tangent  and  secant  curves, 
147-151 

Half-angle,  functions  of,  296 
Halley's  law,  260 
Harmonic  analysis,  336 

curves,  363,  364 

functions,  327 

motion.  Chap.  X,  321  et  seq. 
compound,  334 
Hyperbola,  Chap.  IV  and  XIII. 

asymptotes,  154,  157 

axes,  155,  157 

center,  157 

conjugate,  158 

eccentricity,  406 

foci  and  focal  radii,  404 

latus,  rectum,  407 

parametric  equations,  154 

polar  equation,  409 

rectangular,  54,  153 

symmetrical  equation,  151 

vertices,  157 
Hyperbolic  curves,  51,  54 

sine  and  cosine,  273 

system  of  logarithms,  223 


Hypo  cycloid       and 
choid,  393 


Hypo-tro- 


^  =    V-1,  348 

Identities,    107,    108,    132,    133, 

282-297 
Image  of  curve,  53 
Increasing  function,  58,  152 

progression,  199 
Increment,  logarithmic,  256 
Infinite  discontinuity,  55 

geometrical  progression,  206 
Infinity,  54,  55 
Integral  function,  14,  162 
Intercepts,  40,  41 
Interest,  compound,  205,  256 

curve,  211 
Interpolation,  231 
Intersection  of  loci,  169 
Inverse  of  curve,  130 

of  straight    line    and  circle, 
130 

trigonometric  functions,  132, 
378,  379 
Irrational  function,  14 

numbers,  334 

Lamellar  motion,  82 

Langley's  law,  70 

Latitude  and  longitude  of  a  point, 
26 

Latus  rectum  of  ellipse,  403 
of  hyperbola,  407 
of  parabola,  412 

Law  of  circular  functions,  126 
of  complex  numbers,  350 
of  compound  interest,  256 
of  exponential  function,  267 
of  power  function,  76 
of   sines,    cosines,   and    tan- 
gents, 301-303 

Lead  or  lag,  330,  368 


INDEX 


487 


Legitimate  transformations,  167 

Lemniscate,  387,  389 

Limit,  150 

Limiting  lines  of  ellipse,  146 

Loci,  Chap.  XII,  381  el  seq. 

defined  by  focal  radii,  386 

Theorems  on,  57,  60,  80,  129, 
242,  292 
Locus  of  points,  36 

of  equation,  36 
Logarithmic      and      exponential 
functions.   Chap.   VIII, 
214  et  seq. 

coordinate  paper,  267-274 

curves,  237-244 

double  scale,  245-265 

functions,  219,  223 

increment    and    decrement, 
256,  258,  259,  277 

tables,  229-233 
Logarithm    of   a    number,     216, 

223 
Logarithms,  common,  223 

graph,  216-217 

properties  of,  226-229 

systems  of,  223 

Mantissa,  229 

Mean,  arithmetical,  198 

geometrical,  202 

harmonical,  212 

progressive,  194 
Modulus    of    complex    number, 
356 

of  decay,  259,  277 

of  logarithmic  system,  242 
Motion,  circular,  99 

compound  harmonic,  334 

connecting  rod,  338 

elliptic,  325,  382 

shearing,  81 

S.  H.  M.,  321  et  sea. 


Naperian  base,  220,  223,  228,  257 

system  of  logs.,  223 
Napier,  John,  214 
Natural    system    of    logarithms, 

223 
Negative  angle,  96 

functions  of,  115 
Newton's  law,  260 
Node,  113 
Normal,  130 

equation  of  line,  130,  423 

to  ellipse,  429 

to  parabola,  419 

Oblique  triangles,  300-315 
Odd  functions,  115 
Operators,  344 
Ordinate  of  point,  26 
Origin,  26 

at  vertex,  145,  413 
Orthogonal  systems,  119 
Orthographic  projection,  61,  117, 
137,  158,  243 

Paper,  logarithmic,  268  et  seq. 
polar,  118  et  seq. 
rectangular,  26  et  seq. 
semi-log,  251,  261  et  seq. 
Parabola,  50,  411 
cubical,  50 
polar  equation,  412 
properties  of,  419 
semi-cubical,  50 
Parabolic  curves,  47  et  seq.  267 
Parameter,  140,  381 
Parametric  equations,   140,  381 
of  cycloid,  391 
of  ellipse,  140 
of  hyperbola  154,  155 
Pascal's  triangle,  188,  189 
Periodic     functions     {see     trig.- 
fens.),  30,  113,  360 


488 


INDEX 


Period  of  S.  H.  M.,  322 

of  simple  pendulum,  325 
of  uniform  circular  motion, 

99 
of  wave,  328 
Permutations,  183,  184 

and  combinations,  Chap.  VI, 
182   et  seq. 
Phase  angle,  322,  326,  330 
Plane  triangles,  300-315 
Polar  coordinates,  118,  433 

diagrams   of  periodic   func- 
tions, 120,  298,  360 
equation  of  ellipse,  409 
of  hyperbola,  409 
of  parabola,  412 
of  straight  line,  129 
form  of  complex  number,  358 
relation  to  rectangular,  131, 
433 
Polynomial,  162 

Positive  and  negative  angle,  96, 
115 
coordinates,  26 
side  of  line,  427 
Power  function,  46  et  seq. 

compared    with   exponen- 
tial, 265 
law  of,  76,  77 
practical  graph,  73 
variation  of,  57 
Probability  curve,  197 
Products,  special,  452 
Progressions,  Chap.    VII,  198  et 
seq. 
arithmetical,  198-201,215 
decreasing,  206 
geometrical,  202-210,  215 
harmonical,  211,  212 
Progressive  mean,  194 
Projection,  orthographic,  61,  117, 
137,  158,  243 


Proportionality  factor,  64 

Quadrants,  26 
Quadratic  equations,  457 

systems  of  equations,  171 
Questionable       transformations, 
167 

Radian  unit  of  measure,  97,  98 
Radicals,  reduction  of,  465 
Radius  vector,  118 
Ratio  definition  of  conies,  413 
Rational  formulas,  71 

functions,  14,  162 

numbers,  354 

^   ,.      .sin  a;          tan  x 
Ratio  of and 148, 

X  X  ' 

Rectangular  coords,  (see  Coordi- 
nates), Chap.  II,  26 
et  seq. 

Reflection  of  curve,  53 

Reflector,  87 

Remainder  theorem,  162 

Reversors,  346 

Right  angle  system,  97 

Root  of  any  complex  number,  377 
of  equation,  85 
of  function,  85,  163 
of  unity,  376 

Rotation  of  locus,  78 

polar  coordinates,  127-129 
rectangular,  433-435 
of  rigid  body,  78 

Scalar  numbers,  343 
Scale,  1,  3 

algebraic,  3,  342 

functions,  20 
arithmetical,  3,  342 
double,  5  et  seq. 

logarithmic,  245-265 
uniform,  2 


INDEX 


489 


Scientific  laws  and  formulas,   65 

et  seq. 
Seiche,  332  et  seq. 
S-formulas,  305 
Semi-cubical  parabola,  50 
Semi-logarithmic  paper,  251-261 
Series,  (see  progressions),  19 
Shearing  motion,  82  et  seq. 
Shear  of  circle,  441 

of  ellipse,  436,  441 
of  hyperbola,  441 
of  parabola,  441 
of  straight  line,  81  et  seq. 
Simple  harmonic  function,  327 

motion,  Chap.  X,  321  el 
seq 
pendulum,  63,  195,  325 
Sine,  100 

law,  301 
Sinusoid,  112,  113 
Sinusoidal    varying    magnitude, 
365 
wave,  326 
Slide  rule,  149  et  seq. 
Slope  of  Ime,  39 

of  curve,  40,  113 
Stationary  waves,  332 
Statistical  graphs,  27 
Straight  line,  40,  130,  423,  430 
Strain,  78 
Sub-normal,  419 
Sub-tangent,  240,  419 
Supplementary  angles.  111 
Surds,  reduction  of,  465 
Symmetrical  equation  of  ellipse, 
138 
of  hyperbola,  157* 
systems  of  equations,  174 
Symmetry,  51,  57 

with  respect  to  point,  51 
to  line,  51 
to  curve,  57 


Tables,  damped  vibrations,  279, 
280 
logarithms,  231 
material  in  concrete,  44 
natural  trig,  functions,  104, 

123 
powers,  49,  50 
of  "e,"  241 
Tangent,  100 
graph,  147 
law,  303 

to  cu-cle,  422,  427 
to  curve,  239 
to  ellipse,  428,  429 
to  parabola,  418 
Theorems,  binomial,  189  et  seq. 
factor,  163 
remainder,  162 
functions  of   composite  an- 
gles, 292 
on  loci,  57,  60,  80,  129,  242, 
292 
Transformations,  legitimate  and 

questionable,  167 
Translation,  78,  79,  424 
of  any  locus,  79,  80 
of  point,  424 
of  rigid  body,  78 
Transverse  axis,  157 
Triangle  of  reference,  100,  106 
Triangles,  solution  of,  123,  300- 
315 
oblique,  300-315 
right,  123-125 
Trigonometric  curves,   112,   120, 
147-151,  298 
functions,  100  et  seq. 
Trochoid,  393 
Trochoidal  v/ave,  331 
Trough  of  sinusoid,  113 

Uniform  circular  motion,  99 


490 


INDEX 


Unit,  change  of,  62,  73  et  seq. 
of  angular  measure,  97 


263 


Variables  and  constants,  13 

and  functions  of  variables. 
Chap.  I 
Variation,  63 

of  power  function,  57 
Vector,  118,  357 

radius,  118 
Vectorial  angle,  100,  118 
Velocity,  angular,  99,  322 

of  wave,  329 


Versors,  347 
Vertices  of  ellipse,  138 
of  hyperbola,  157 
Vibrations,  damped,  276 

Waves,  Chap.  X.,  326  e^ 
compound,  334 
length  of,  327 
sinusoidal,  326  el  seq. 
stationary,  331 
trochoidal,  331 

Zero  of  function,  85 


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